Почетна
Случајна
У близини
Подешавања
О Википедији
Одрицање одговорности
Search
Списак интеграла тригонометријских функција
списак на Викимедији
Садржај
Интеграли синусних функција
Интеграли косинусних функција
Интеграли тангенсних функција
Интеграли секанс функција
Интеграли косенканс функција
Интеграли котангенс функција
Интеграли синусних и косинусних функција
Интеграли синусних и тангенсних функција
Интеграли косинусних и тангенсних функција
Интеграли синусних и котангенсних функција
Интеграли косинусних и котангенсних функција
Интеграли са симетричним границама
Литература
Интеграли
синусних
функција
∫
sin
a
x
d
x
=
−
1
a
cos
a
x
+
C
{\displaystyle \int \sin ax\;\mathrm {d} x=-{\frac {1}{a}}\cos ax+C\,\!}
∫
sin
2
a
x
d
x
=
x
2
−
1
4
a
sin
2
a
x
+
C
=
x
2
−
1
2
a
sin
a
x
cos
a
x
+
C
{\displaystyle \int \sin ^{2}{ax}\;\mathrm {d} x={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}-{\frac {1}{2a}}\sin ax\cos ax+C\!}
∫
sin
3
a
x
d
x
=
cos
3
a
x
12
a
−
3
cos
a
x
4
a
+
C
{\displaystyle \int \sin ^{3}{ax}\;\mathrm {d} x={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C\!}
∫
x
sin
2
a
x
d
x
=
x
2
4
−
x
4
a
sin
2
a
x
−
1
8
a
2
cos
2
a
x
+
C
{\displaystyle \int x\sin ^{2}{ax}\;\mathrm {d} x={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+C\!}
∫
x
2
sin
2
a
x
d
x
=
x
3
6
−
(
x
2
4
a
−
1
8
a
3
)
sin
2
a
x
−
x
4
a
2
cos
2
a
x
+
C
{\displaystyle \int x^{2}\sin ^{2}{ax}\;\mathrm {d} x={\frac {x^{3}}{6}}-\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax-{\frac {x}{4a^{2}}}\cos 2ax+C\!}
∫
sin
b
1
x
sin
b
2
x
d
x
=
sin
(
(
b
2
−
b
1
)
x
)
2
(
b
2
−
b
1
)
−
sin
(
(
b
1
+
b
2
)
x
)
2
(
b
1
+
b
2
)
+
C
(for
|
b
1
|
≠
|
b
2
|
)
{\displaystyle \int \sin b_{1}x\sin b_{2}x\;\mathrm {d} x={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+C\qquad {\mbox{(for }}|b_{1}|\neq |b_{2}|{\mbox{)}}\,\!}
∫
sin
n
a
x
d
x
=
−
sin
n
−
1
a
x
cos
a
x
n
a
+
n
−
1
n
∫
sin
n
−
2
a
x
d
x
(for
n
>
2
)
{\displaystyle \int \sin ^{n}{ax}\;\mathrm {d} x=-{\frac {\sin ^{n-1}ax\cos ax}{na}}+{\frac {n-1}{n}}\int \sin ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n>2{\mbox{)}}\,\!}
∫
d
x
sin
a
x
=
1
a
ln
|
tan
a
x
2
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ax}}={\frac {1}{a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}
∫
d
x
sin
n
a
x
=
cos
a
x
a
(
1
−
n
)
sin
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
sin
n
−
2
a
x
(for
n
>
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{n}ax}}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}
∫
x
sin
a
x
d
x
=
sin
a
x
a
2
−
x
cos
a
x
a
+
C
{\displaystyle \int x\sin ax\;\mathrm {d} x={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+C\,\!}
∫
x
n
sin
a
x
d
x
=
−
x
n
a
cos
a
x
+
n
a
∫
x
n
−
1
cos
a
x
d
x
=
∑
k
=
0
2
k
≤
n
(
−
1
)
k
+
1
x
n
−
2
k
a
1
+
2
k
n
!
(
n
−
2
k
)
!
cos
a
x
+
∑
k
=
0
2
k
+
1
≤
n
(
−
1
)
k
x
n
−
1
−
2
k
a
2
+
2
k
n
!
(
n
−
2
k
−
1
)
!
sin
a
x
(for
n
>
0
)
{\displaystyle \int x^{n}\sin ax\;\mathrm {d} x=-{\frac {x^{n}}{a}}\cos ax+{\frac {n}{a}}\int x^{n-1}\cos ax\;\mathrm {d} x=\sum _{k=0}^{2k\leq n}(-1)^{k+1}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\cos ax+\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-1-2k}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\sin ax\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}
∫
−
a
2
a
2
x
2
sin
2
n
π
x
a
d
x
=
a
3
(
n
2
π
2
−
6
)
24
n
2
π
2
(for
n
=
2
,
4
,
6...
)
{\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\;\mathrm {d} x={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=2,4,6...{\mbox{)}}\,\!}
∫
sin
a
x
x
d
x
=
∑
n
=
0
∞
(
−
1
)
n
(
a
x
)
2
n
+
1
(
2
n
+
1
)
⋅
(
2
n
+
1
)
!
+
C
{\displaystyle \int {\frac {\sin ax}{x}}\mathrm {d} x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+C\,\!}
∫
sin
a
x
x
n
d
x
=
−
sin
a
x
(
n
−
1
)
x
n
−
1
+
a
n
−
1
∫
cos
a
x
x
n
−
1
d
x
{\displaystyle \int {\frac {\sin ax}{x^{n}}}\mathrm {d} x=-{\frac {\sin ax}{(n-1)x^{n-1}}}+{\frac {a}{n-1}}\int {\frac {\cos ax}{x^{n-1}}}\mathrm {d} x\,\!}
∫
d
x
1
±
sin
a
x
=
1
a
tan
(
a
x
2
∓
π
4
)
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+C}
∫
x
d
x
1
+
sin
a
x
=
x
a
tan
(
a
x
2
−
π
4
)
+
2
a
2
ln
|
cos
(
a
x
2
−
π
4
)
|
+
C
{\displaystyle \int {\frac {x\;\mathrm {d} x}{1+\sin ax}}={\frac {x}{a}}\tan \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{a^{2}}}\ln \left|\cos \left({\frac {ax}{2}}-{\frac {\pi }{4}}\right)\right|+C}
∫
x
d
x
1
−
sin
a
x
=
x
a
cot
(
π
4
−
a
x
2
)
+
2
a
2
ln
|
sin
(
π
4
−
a
x
2
)
|
+
C
{\displaystyle \int {\frac {x\;\mathrm {d} x}{1-\sin ax}}={\frac {x}{a}}\cot \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)+{\frac {2}{a^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {ax}{2}}\right)\right|+C}
∫
sin
a
x
d
x
1
±
sin
a
x
=
±
x
+
1
a
tan
(
π
4
∓
a
x
2
)
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{1\pm \sin ax}}=\pm x+{\frac {1}{a}}\tan \left({\frac {\pi }{4}}\mp {\frac {ax}{2}}\right)+C}
Интеграли
косинусних
функција
∫
cos
a
x
d
x
=
1
a
sin
a
x
+
C
{\displaystyle \int \cos ax\;\mathrm {d} x={\frac {1}{a}}\sin ax+C\,\!}
∫
cos
2
a
x
d
x
=
x
2
+
1
4
a
sin
2
a
x
+
C
=
x
2
+
1
2
a
sin
a
x
cos
a
x
+
C
{\displaystyle \int \cos ^{2}{ax}\;\mathrm {d} x={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C={\frac {x}{2}}+{\frac {1}{2a}}\sin ax\cos ax+C\!}
∫
cos
n
a
x
d
x
=
cos
n
−
1
a
x
sin
a
x
n
a
+
n
−
1
n
∫
cos
n
−
2
a
x
d
x
(for
n
>
0
)
{\displaystyle \int \cos ^{n}ax\;\mathrm {d} x={\frac {\cos ^{n-1}ax\sin ax}{na}}+{\frac {n-1}{n}}\int \cos ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}
∫
x
cos
a
x
d
x
=
cos
a
x
a
2
+
x
sin
a
x
a
+
C
{\displaystyle \int x\cos ax\;\mathrm {d} x={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+C\,\!}
∫
x
2
cos
2
a
x
d
x
=
x
3
6
+
(
x
2
4
a
−
1
8
a
3
)
sin
2
a
x
+
x
4
a
2
cos
2
a
x
+
C
{\displaystyle \int x^{2}\cos ^{2}{ax}\;\mathrm {d} x={\frac {x^{3}}{6}}+\left({\frac {x^{2}}{4a}}-{\frac {1}{8a^{3}}}\right)\sin 2ax+{\frac {x}{4a^{2}}}\cos 2ax+C\!}
∫
x
n
cos
a
x
d
x
=
x
n
sin
a
x
a
−
n
a
∫
x
n
−
1
sin
a
x
d
x
=
∑
k
=
0
2
k
+
1
≤
n
(
−
1
)
k
x
n
−
2
k
−
1
a
2
+
2
k
n
!
(
n
−
2
k
−
1
)
!
cos
a
x
+
∑
k
=
0
2
k
≤
n
(
−
1
)
k
x
n
−
2
k
a
1
+
2
k
n
!
(
n
−
2
k
)
!
sin
a
x
{\displaystyle \int x^{n}\cos ax\;\mathrm {d} x={\frac {x^{n}\sin ax}{a}}-{\frac {n}{a}}\int x^{n-1}\sin ax\;\mathrm {d} x\,=\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-2k-1}}{a^{2+2k}}}{\frac {n!}{(n-2k-1)!}}\cos ax+\sum _{k=0}^{2k\leq n}(-1)^{k}{\frac {x^{n-2k}}{a^{1+2k}}}{\frac {n!}{(n-2k)!}}\sin ax\!}
∫
cos
a
x
x
d
x
=
ln
|
a
x
|
+
∑
k
=
1
∞
(
−
1
)
k
(
a
x
)
2
k
2
k
⋅
(
2
k
)
!
+
C
{\displaystyle \int {\frac {\cos ax}{x}}\mathrm {d} x=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+C\,\!}
∫
cos
a
x
x
n
d
x
=
−
cos
a
x
(
n
−
1
)
x
n
−
1
−
a
n
−
1
∫
sin
a
x
x
n
−
1
d
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\cos ax}{x^{n}}}\mathrm {d} x=-{\frac {\cos ax}{(n-1)x^{n-1}}}-{\frac {a}{n-1}}\int {\frac {\sin ax}{x^{n-1}}}\mathrm {d} x\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
cos
a
x
=
1
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
∫
d
x
cos
n
a
x
=
sin
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
+
n
−
2
n
−
1
∫
d
x
cos
n
−
2
a
x
(for
n
>
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}
∫
d
x
1
+
cos
a
x
=
1
a
tan
a
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}
∫
d
x
1
−
cos
a
x
=
−
1
a
cot
a
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}
∫
x
d
x
1
+
cos
a
x
=
x
a
tan
a
x
2
+
2
a
2
ln
|
cos
a
x
2
|
+
C
{\displaystyle \int {\frac {x\;\mathrm {d} x}{1+\cos ax}}={\frac {x}{a}}\tan {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\cos {\frac {ax}{2}}\right|+C}
∫
x
d
x
1
−
cos
a
x
=
−
x
a
cot
a
x
2
+
2
a
2
ln
|
sin
a
x
2
|
+
C
{\displaystyle \int {\frac {x\;\mathrm {d} x}{1-\cos ax}}=-{\frac {x}{a}}\cot {\frac {ax}{2}}+{\frac {2}{a^{2}}}\ln \left|\sin {\frac {ax}{2}}\right|+C}
∫
cos
a
x
d
x
1
+
cos
a
x
=
x
−
1
a
tan
a
x
2
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+C\,\!}
∫
cos
a
x
d
x
1
−
cos
a
x
=
−
x
−
1
a
cot
a
x
2
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+C\,\!}
∫
cos
a
1
x
cos
a
2
x
d
x
=
sin
(
a
2
−
a
1
)
x
2
(
a
2
−
a
1
)
+
sin
(
a
2
+
a
1
)
x
2
(
a
2
+
a
1
)
+
C
(for
|
a
1
|
≠
|
a
2
|
)
{\displaystyle \int \cos a_{1}x\cos a_{2}x\;\mathrm {d} x={\frac {\sin(a_{2}-a_{1})x}{2(a_{2}-a_{1})}}+{\frac {\sin(a_{2}+a_{1})x}{2(a_{2}+a_{1})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}
Интеграли
тангенсних
функција
∫
tan
a
x
d
x
=
−
1
a
ln
|
cos
a
x
|
+
C
=
1
a
ln
|
sec
a
x
|
+
C
{\displaystyle \int \tan ax\;\mathrm {d} x=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+C\,\!}
∫
tan
n
a
x
d
x
=
1
a
(
n
−
1
)
tan
n
−
1
a
x
−
∫
tan
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \tan ^{n}ax\;\mathrm {d} x={\frac {1}{a(n-1)}}\tan ^{n-1}ax-\int \tan ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
q
tan
a
x
+
p
=
1
p
2
+
q
2
(
p
x
+
q
a
ln
|
q
sin
a
x
+
p
cos
a
x
|
)
+
C
(for
p
2
+
q
2
≠
0
)
{\displaystyle \int {\frac {\mathrm {d} x}{q\tan ax+p}}={\frac {1}{p^{2}+q^{2}}}(px+{\frac {q}{a}}\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(for }}p^{2}+q^{2}\neq 0{\mbox{)}}\,\!}
∫
d
x
tan
a
x
+
1
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\tan ax+1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}
∫
d
x
tan
a
x
−
1
=
−
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\tan ax-1}}=-{\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}
∫
tan
a
x
d
x
tan
a
x
+
1
=
x
2
−
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\tan ax\;\mathrm {d} x}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+C\,\!}
∫
tan
a
x
d
x
tan
a
x
−
1
=
x
2
+
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\tan ax\;\mathrm {d} x}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+C\,\!}
Интеграли
секанс
функција
∫
sec
a
x
d
x
=
1
a
ln
|
sec
a
x
+
tan
a
x
|
+
C
{\displaystyle \int \sec {ax}\,\mathrm {d} x={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+C}
∫
sec
2
x
d
x
=
tan
x
+
C
{\displaystyle \int \sec ^{2}{x}\,\mathrm {d} x=\tan {x}+C}
∫
sec
n
a
x
d
x
=
sec
n
−
2
a
x
tan
a
x
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
sec
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \sec ^{n}{ax}\,\mathrm {d} x={\frac {\sec ^{n-2}{ax}\tan {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{ax}\,\mathrm {d} x\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}
∫
sec
n
x
d
x
=
sec
n
−
2
x
tan
x
n
−
1
+
n
−
2
n
−
1
∫
sec
n
−
2
x
d
x
{\displaystyle \int \sec ^{n}{x}\,\mathrm {d} x={\frac {\sec ^{n-2}{x}\tan {x}}{n-1}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{x}\,\mathrm {d} x}
∫
d
x
sec
x
+
1
=
x
−
tan
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+C}
∫
d
x
sec
x
−
1
=
−
x
−
cot
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sec {x}-1}}=-x-\cot {\frac {x}{2}}+C}
Интеграли косенканс функција
∫
csc
a
x
d
x
=
1
a
ln
|
csc
a
x
−
cot
a
x
|
+
C
{\displaystyle \int \csc {ax}\,\mathrm {d} x={\frac {1}{a}}\ln {\left|\csc {ax}-\cot {ax}\right|}+C}
∫
csc
2
x
d
x
=
−
cot
x
+
C
{\displaystyle \int \csc ^{2}{x}\,\mathrm {d} x=-\cot {x}+C}
∫
csc
n
a
x
d
x
=
−
csc
n
−
1
a
x
cos
a
x
a
(
n
−
1
)
+
n
−
2
n
−
1
∫
csc
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \csc ^{n}{ax}\,\mathrm {d} x=-{\frac {\csc ^{n-1}{ax}\cos {ax}}{a(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{ax}\,\mathrm {d} x\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
csc
x
+
1
=
x
−
2
sin
x
2
cos
x
2
+
sin
x
2
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\csc {x}+1}}=x-{\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}}+C}
∫
d
x
csc
x
−
1
=
2
sin
x
2
cos
x
2
−
sin
x
2
−
x
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\csc {x}-1}}={\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}-x+C}
Интеграли
котангенс
функција
∫
cot
a
x
d
x
=
1
a
ln
|
sin
a
x
|
+
C
{\displaystyle \int \cot ax\;\mathrm {d} x={\frac {1}{a}}\ln |\sin ax|+C\,\!}
∫
cot
n
a
x
d
x
=
−
1
a
(
n
−
1
)
cot
n
−
1
a
x
−
∫
cot
n
−
2
a
x
d
x
(for
n
≠
1
)
{\displaystyle \int \cot ^{n}ax\;\mathrm {d} x=-{\frac {1}{a(n-1)}}\cot ^{n-1}ax-\int \cot ^{n-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
1
+
cot
a
x
=
∫
tan
a
x
d
x
tan
a
x
+
1
{\displaystyle \int {\frac {\mathrm {d} x}{1+\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax+1}}\,\!}
∫
d
x
1
−
cot
a
x
=
∫
tan
a
x
d
x
tan
a
x
−
1
{\displaystyle \int {\frac {\mathrm {d} x}{1-\cot ax}}=\int {\frac {\tan ax\;\mathrm {d} x}{\tan ax-1}}\,\!}
Интеграли
синусних
и
косинусних
функција
∫
d
x
cos
a
x
±
sin
a
x
=
1
a
2
ln
|
tan
(
a
x
2
±
π
8
)
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\cos ax\pm \sin ax}}={\frac {1}{a{\sqrt {2}}}}\ln \left|\tan \left({\frac {ax}{2}}\pm {\frac {\pi }{8}}\right)\right|+C}
∫
d
x
(
cos
a
x
±
sin
a
x
)
2
=
1
2
a
tan
(
a
x
∓
π
4
)
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+C}
∫
d
x
(
cos
x
+
sin
x
)
n
=
1
n
−
1
(
sin
x
−
cos
x
(
cos
x
+
sin
x
)
n
−
1
−
2
(
n
−
2
)
∫
d
x
(
cos
x
+
sin
x
)
n
−
2
)
{\displaystyle \int {\frac {\mathrm {d} x}{(\cos x+\sin x)^{n}}}={\frac {1}{n-1}}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}}}-2(n-2)\int {\frac {\mathrm {d} x}{(\cos x+\sin x)^{n-2}}}\right)}
∫
cos
a
x
d
x
cos
a
x
+
sin
a
x
=
x
2
+
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
∫
cos
a
x
d
x
cos
a
x
−
sin
a
x
=
x
2
−
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}
∫
sin
a
x
d
x
cos
a
x
+
sin
a
x
=
x
2
−
1
2
a
ln
|
sin
a
x
+
cos
a
x
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+C}
∫
sin
a
x
d
x
cos
a
x
−
sin
a
x
=
−
x
2
−
1
2
a
ln
|
sin
a
x
−
cos
a
x
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+C}
∫
cos
a
x
d
x
sin
a
x
(
1
+
cos
a
x
)
=
−
1
4
a
tan
2
a
x
2
+
1
2
a
ln
|
tan
a
x
2
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\sin ax(1+\cos ax)}}=-{\frac {1}{4a}}\tan ^{2}{\frac {ax}{2}}+{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}
∫
cos
a
x
d
x
sin
a
x
(
1
−
cos
a
x
)
=
−
1
4
a
cot
2
a
x
2
−
1
2
a
ln
|
tan
a
x
2
|
+
C
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\sin ax(1-\cos ax)}}=-{\frac {1}{4a}}\cot ^{2}{\frac {ax}{2}}-{\frac {1}{2a}}\ln \left|\tan {\frac {ax}{2}}\right|+C}
∫
sin
a
x
d
x
cos
a
x
(
1
+
sin
a
x
)
=
1
4
a
cot
2
(
a
x
2
+
π
4
)
+
1
2
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ax(1+\sin ax)}}={\frac {1}{4a}}\cot ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
∫
sin
a
x
d
x
cos
a
x
(
1
−
sin
a
x
)
=
1
4
a
tan
2
(
a
x
2
+
π
4
)
−
1
2
a
ln
|
tan
(
a
x
2
+
π
4
)
|
+
C
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ax(1-\sin ax)}}={\frac {1}{4a}}\tan ^{2}\left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+C}
∫
sin
a
x
cos
a
x
d
x
=
−
1
2
a
cos
2
a
x
+
C
{\displaystyle \int \sin ax\cos ax\;\mathrm {d} x=-{\frac {1}{2a}}\cos ^{2}ax+C\,\!}
∫
sin
a
1
x
cos
a
2
x
d
x
=
−
cos
(
(
a
1
−
a
2
)
x
)
2
(
a
1
−
a
2
)
−
cos
(
(
a
1
+
a
2
)
x
)
2
(
a
1
+
a
2
)
+
C
(for
|
a
1
|
≠
|
a
2
|
)
{\displaystyle \int \sin a_{1}x\cos a_{2}x\;\mathrm {d} x=-{\frac {\cos((a_{1}-a_{2})x)}{2(a_{1}-a_{2})}}-{\frac {\cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})}}+C\qquad {\mbox{(for }}|a_{1}|\neq |a_{2}|{\mbox{)}}\,\!}
∫
sin
n
a
x
cos
a
x
d
x
=
1
a
(
n
+
1
)
sin
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int \sin ^{n}ax\cos ax\;\mathrm {d} x={\frac {1}{a(n+1)}}\sin ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
∫
sin
a
x
cos
n
a
x
d
x
=
−
1
a
(
n
+
1
)
cos
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int \sin ax\cos ^{n}ax\;\mathrm {d} x=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
∫
sin
n
a
x
cos
m
a
x
d
x
=
−
sin
n
−
1
a
x
cos
m
+
1
a
x
a
(
n
+
m
)
+
n
−
1
n
+
m
∫
sin
n
−
2
a
x
cos
m
a
x
d
x
(for
m
,
n
>
0
)
{\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;\mathrm {d} x=-{\frac {\sin ^{n-1}ax\cos ^{m+1}ax}{a(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}ax\cos ^{m}ax\;\mathrm {d} x\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}
also:
∫
sin
n
a
x
cos
m
a
x
d
x
=
sin
n
+
1
a
x
cos
m
−
1
a
x
a
(
n
+
m
)
+
m
−
1
n
+
m
∫
sin
n
a
x
cos
m
−
2
a
x
d
x
(for
m
,
n
>
0
)
{\displaystyle \int \sin ^{n}ax\cos ^{m}ax\;\mathrm {d} x={\frac {\sin ^{n+1}ax\cos ^{m-1}ax}{a(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}ax\cos ^{m-2}ax\;\mathrm {d} x\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}
∫
d
x
sin
a
x
cos
a
x
=
1
a
ln
|
tan
a
x
|
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ax\cos ax}}={\frac {1}{a}}\ln \left|\tan ax\right|+C}
∫
d
x
sin
a
x
cos
n
a
x
=
1
a
(
n
−
1
)
cos
n
−
1
a
x
+
∫
d
x
sin
a
x
cos
n
−
2
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ax\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+\int {\frac {\mathrm {d} x}{\sin ax\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
d
x
sin
n
a
x
cos
a
x
=
−
1
a
(
n
−
1
)
sin
n
−
1
a
x
+
∫
d
x
sin
n
−
2
a
x
cos
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{n}ax\cos ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
a
x
d
x
cos
n
a
x
=
1
a
(
n
−
1
)
cos
n
−
1
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\sin ax\;\mathrm {d} x}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
2
a
x
d
x
cos
a
x
=
−
1
a
sin
a
x
+
1
a
ln
|
tan
(
π
4
+
a
x
2
)
|
+
C
{\displaystyle \int {\frac {\sin ^{2}ax\;\mathrm {d} x}{\cos ax}}=-{\frac {1}{a}}\sin ax+{\frac {1}{a}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {ax}{2}}\right)\right|+C}
∫
sin
2
a
x
d
x
cos
n
a
x
=
sin
a
x
a
(
n
−
1
)
cos
n
−
1
a
x
−
1
n
−
1
∫
d
x
cos
n
−
2
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\sin ^{2}ax\;\mathrm {d} x}{\cos ^{n}ax}}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}}-{\frac {1}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
n
a
x
d
x
cos
a
x
=
−
sin
n
−
1
a
x
a
(
n
−
1
)
+
∫
sin
n
−
2
a
x
d
x
cos
a
x
(for
n
≠
1
)
{\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ax}}=-{\frac {\sin ^{n-1}ax}{a(n-1)}}+\int {\frac {\sin ^{n-2}ax\;\mathrm {d} x}{\cos ax}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
sin
n
a
x
d
x
cos
m
a
x
=
sin
n
+
1
a
x
a
(
m
−
1
)
cos
m
−
1
a
x
−
n
−
m
+
2
m
−
1
∫
sin
n
a
x
d
x
cos
m
−
2
a
x
(for
m
≠
1
)
{\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m}ax}}={\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}
also:
∫
sin
n
a
x
d
x
cos
m
a
x
=
−
sin
n
−
1
a
x
a
(
n
−
m
)
cos
m
−
1
a
x
+
n
−
1
n
−
m
∫
sin
n
−
2
a
x
d
x
cos
m
a
x
(for
m
≠
n
)
{\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m}ax}}=-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}ax\;\mathrm {d} x}{\cos ^{m}ax}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}
also:
∫
sin
n
a
x
d
x
cos
m
a
x
=
sin
n
−
1
a
x
a
(
m
−
1
)
cos
m
−
1
a
x
−
n
−
1
m
−
1
∫
sin
n
−
2
a
x
d
x
cos
m
−
2
a
x
(for
m
≠
1
)
{\displaystyle \int {\frac {\sin ^{n}ax\;\mathrm {d} x}{\cos ^{m}ax}}={\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-2}ax\;\mathrm {d} x}{\cos ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}
∫
cos
a
x
d
x
sin
n
a
x
=
−
1
a
(
n
−
1
)
sin
n
−
1
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\cos ax\;\mathrm {d} x}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
∫
cos
2
a
x
d
x
sin
a
x
=
1
a
(
cos
a
x
+
ln
|
tan
a
x
2
|
)
+
C
{\displaystyle \int {\frac {\cos ^{2}ax\;\mathrm {d} x}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+C}
∫
cos
2
a
x
d
x
sin
n
a
x
=
−
1
n
−
1
(
cos
a
x
a
sin
n
−
1
a
x
)
+
∫
d
x
sin
n
−
2
a
x
)
(for
n
≠
1
)
{\displaystyle \int {\frac {\cos ^{2}ax\;\mathrm {d} x}{\sin ^{n}ax}}=-{\frac {1}{n-1}}\left({\frac {\cos ax}{a\sin ^{n-1}ax)}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}ax}}\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
cos
n
a
x
d
x
sin
m
a
x
=
−
cos
n
+
1
a
x
a
(
m
−
1
)
sin
m
−
1
a
x
−
n
−
m
−
2
m
−
1
∫
cos
n
a
x
d
x
sin
m
−
2
a
x
(for
m
≠
1
)
{\displaystyle \int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\sin ^{m}ax}}=-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-m-2}{m-1}}\int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\sin ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}
also:
∫
cos
n
a
x
d
x
sin
m
a
x
=
cos
n
−
1
a
x
a
(
n
−
m
)
sin
m
−
1
a
x
+
n
−
1
n
−
m
∫
cos
n
−
2
a
x
d
x
sin
m
a
x
(for
m
≠
n
)
{\displaystyle \int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\sin ^{m}ax}}={\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cos ^{n-2}ax\;\mathrm {d} x}{\sin ^{m}ax}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}
also:
∫
cos
n
a
x
d
x
sin
m
a
x
=
−
cos
n
−
1
a
x
a
(
m
−
1
)
sin
m
−
1
a
x
−
n
−
1
m
−
1
∫
cos
n
−
2
a
x
d
x
sin
m
−
2
a
x
(for
m
≠
1
)
{\displaystyle \int {\frac {\cos ^{n}ax\;\mathrm {d} x}{\sin ^{m}ax}}=-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax}}-{\frac {n-1}{m-1}}\int {\frac {\cos ^{n-2}ax\;\mathrm {d} x}{\sin ^{m-2}ax}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}
Интеграли
синусних
и
тангенсних
функција
∫
sin
a
x
tan
a
x
d
x
=
1
a
(
ln
|
sec
a
x
+
tan
a
x
|
−
sin
a
x
)
+
C
{\displaystyle \int \sin ax\tan ax\;\mathrm {d} x={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+C\,\!}
∫
tan
n
a
x
d
x
sin
2
a
x
=
1
a
(
n
−
1
)
tan
n
−
1
(
a
x
)
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\tan ^{n}ax\;\mathrm {d} x}{\sin ^{2}ax}}={\frac {1}{a(n-1)}}\tan ^{n-1}(ax)+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
Интеграли
косинусних
и
тангенсних
функција
∫
tan
n
a
x
d
x
cos
2
a
x
=
1
a
(
n
+
1
)
tan
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int {\frac {\tan ^{n}ax\;\mathrm {d} x}{\cos ^{2}ax}}={\frac {1}{a(n+1)}}\tan ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
Интеграли
синусних
и
котангенсних
функција
∫
cot
n
a
x
d
x
sin
2
a
x
=
−
1
a
(
n
+
1
)
cot
n
+
1
a
x
+
C
(for
n
≠
−
1
)
{\displaystyle \int {\frac {\cot ^{n}ax\;\mathrm {d} x}{\sin ^{2}ax}}=-{\frac {1}{a(n+1)}}\cot ^{n+1}ax+C\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}
Интеграли
косинусних
и
котангенсних
функција
∫
cot
n
a
x
d
x
cos
2
a
x
=
1
a
(
1
−
n
)
tan
1
−
n
a
x
+
C
(for
n
≠
1
)
{\displaystyle \int {\frac {\cot ^{n}ax\;\mathrm {d} x}{\cos ^{2}ax}}={\frac {1}{a(1-n)}}\tan ^{1-n}ax+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}
Интеграли са симетричним границама
∫
−
c
c
sin
x
d
x
=
0
{\displaystyle \int _{-c}^{c}\sin {x}\;\mathrm {d} x=0\!}
∫
−
c
c
cos
x
d
x
=
2
∫
0
c
cos
x
d
x
=
2
∫
−
c
0
cos
x
d
x
=
2
sin
c
{\displaystyle \int _{-c}^{c}\cos {x}\;\mathrm {d} x=2\int _{0}^{c}\cos {x}\;\mathrm {d} x=2\int _{-c}^{0}\cos {x}\;\mathrm {d} x=2\sin {c}\!}
∫
−
c
c
tan
x
d
x
=
0
{\displaystyle \int _{-c}^{c}\tan {x}\;\mathrm {d} x=0\!}
∫
−
a
2
a
2
x
2
cos
2
n
π
x
a
d
x
=
a
3
(
n
2
π
2
−
6
)
24
n
2
π
2
(for
n
=
1
,
3
,
5...
)
{\displaystyle \int _{-{\frac {a}{2}}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\;\mathrm {d} x={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=1,3,5...{\mbox{)}}\,\!}
Литература
🔥 Top keywords:
Главна страна
Европско првенство у фудбалу 2024.
Вања Маринковић
Стилске фигуре
Европско првенство у фудбалу
Раде Богдановић
Душан Тадић
Марко Арнаутовић
Посебно:Претражи
Душан Влаховић
Килијан Мбапе
Словачка
Фудбалска репрезентација Србије
Србија
Драган Стојковић Пикси
Европско првенство у фудбалу 2020.
Данило I Петровић Његош (књаз)
Светско првенство у фудбалу
Вељко Бирманчевић
Прилог (врста речи)
Светско првенство у фудбалу 2022.
Књижевни родови и врсте
Мирдита, добар дан!
Александар Митровић
Матија Шаркић
Страхиња Павловић
Вук Стефановић Караџић
Џуд Белингам
Развој језика код Срба од 9. до 19. века
Румунија
Дијалекти српског језика
Филип Костић
Фудбалска репрезентација Француске
Ромелу Лукаку
Предраг Рајковић
Предлог
Белгија
Антоан Гризман
Персонификација
