三角函数积分表

三角函数积分表

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以下是部份三角函數的積分表(省略积分常数):

目录

1 积分只有sin的函數

2 积分只有cos的函數

3 积分只有tan的函數

4 积分只有sec的函數

5 积分只有csc的函數

6 积分只有cot的函數

7 积分只有sin和cos的函數

8 积分只有sin和tan的函數

9 积分只有cos和tan的函數

10 积分只有sin和cot的函數

11 积分只有cos和cot的函數

12 积分只有tan和cot的函數

积分只有sin的函數

编辑

sin

c

x

d

x

=

1

c

cos

c

x

{\displaystyle \int \sin cx\;dx=-{\frac {1}{c}}\cos cx\,\!}

sin

n

c

x

d

x

=

1

n

c

sin

n

1

c

x

cos

c

x

+

n

1

n

sin

n

2

c

x

d

x

(

{\displaystyle \int \sin ^{n}cx\;dx=-{\frac {1}{nc}}\sin ^{n-1}cx\cos cx+{\frac {n-1}{n}}\int \sin ^{n-2}cx\;dx\qquad (}

其中

n

>

0

)

{\displaystyle n>0\,\!)}

1

sin

x

d

x

=

cvs

x

d

x

=

2

cos

x

2

+

sin

x

2

cos

x

2

sin

x

2

cvs

x

(

=

2

1

+

sin

x

)

{\displaystyle \int {\sqrt {1-\sin {x}}}\,dx=\int {\sqrt {\operatorname {cvs} {x}}}\,dx=2{\frac {\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}{\sqrt {\operatorname {cvs} {x}}}(=2{\sqrt {1+\sin x}})}

(其中

cvs

x

{\displaystyle \operatorname {cvs} {x}}

是 餘矢(Coversine)函數(參閱正矢(versine)函數))

x

sin

c

x

d

x

=

sin

c

x

c

2

x

cos

c

x

c

{\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}}}-{\frac {x\cos cx}{c}}\,\!}

x

n

sin

c

x

d

x

=

x

n

c

cos

c

x

+

n

c

x

n

1

cos

c

x

d

x

(

{\displaystyle \int x^{n}\sin cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad (}

其中

n

>

0

)

{\displaystyle n>0\,\!)}

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

(

{\displaystyle \int _{-{\frac {a}{2}}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad (}

其中

n

=

2

,

4

,

6...

)

{\displaystyle n=2,4,6...\,\!)}

sin

c

x

x

d

x

=

i

=

0

(

1

)

i

(

c

x

)

2

i

+

1

(

2

i

+

1

)

(

2

i

+

1

)

!

{\displaystyle \int {\frac {\sin cx}{x}}dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}\,\!}

sin

c

x

x

n

d

x

=

sin

c

x

(

n

1

)

x

n

1

+

c

n

1

cos

c

x

x

n

1

d

x

{\displaystyle \int {\frac {\sin cx}{x^{n}}}dx=-{\frac {\sin cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos cx}{x^{n-1}}}dx\,\!}

d

x

sin

c

x

=

1

c

ln

|

tan

c

x

2

|

{\displaystyle \int {\frac {dx}{\sin cx}}={\frac {1}{c}}\ln \left|\tan {\frac {cx}{2}}\right|}

d

x

sin

n

c

x

=

cos

c

x

c

(

1

n

)

sin

n

1

c

x

+

n

2

n

1

d

x

sin

n

2

c

x

(

{\displaystyle \int {\frac {dx}{\sin ^{n}cx}}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad (}

其中

n

>

1

)

{\displaystyle n>1\,\!)}

d

x

1

±

sin

c

x

=

1

c

tan

(

c

x

2

π

4

)

{\displaystyle \int {\frac {dx}{1\pm \sin cx}}={\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}

x

d

x

1

+

sin

c

x

=

x

c

tan

(

c

x

2

π

4

)

+

2

c

2

ln

|

cos

(

c

x

2

π

4

)

|

{\displaystyle \int {\frac {x\;dx}{1+\sin cx}}={\frac {x}{c}}\tan \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{c^{2}}}\ln \left|\cos \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)\right|}

x

d

x

1

sin

c

x

=

x

c

cot

(

π

4

c

x

2

)

+

2

c

2

ln

|

sin

(

π

4

c

x

2

)

|

{\displaystyle \int {\frac {x\;dx}{1-\sin cx}}={\frac {x}{c}}\cot \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}

sin

c

x

d

x

1

±

sin

c

x

=

±

x

+

1

c

tan

(

π

4

c

x

2

)

{\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}}=\pm x+{\frac {1}{c}}\tan \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}

sin

c

1

x

sin

c

2

x

d

x

=

sin

(

c

1

c

2

)

x

2

(

c

1

c

2

)

sin

(

c

1

+

c

2

)

x

2

(

c

1

+

c

2

)

(

{\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad (}

其中

|

c

1

|

|

c

2

|

)

{\displaystyle |c_{1}|\neq |c_{2}|\,\!)}

积分只有cos的函數

编辑

cos

c

x

d

x

=

1

c

sin

c

x

{\displaystyle \int \cos cx\;dx={\frac {1}{c}}\sin cx\,\!}

cos

n

c

x

d

x

=

1

n

c

cos

n

1

c

x

sin

c

x

+

n

1

n

cos

n

2

c

x

d

x

(

n

>

0

)

{\displaystyle \int \cos ^{n}cx\;dx={\frac {1}{nc}}\cos ^{n-1}cx\sin cx+{\frac {n-1}{n}}\int \cos ^{n-2}cx\;dx\qquad {\mbox{(}}n>0{\mbox{)}}\,\!}

x

cos

c

x

d

x

=

cos

c

x

c

2

+

x

sin

c

x

c

{\displaystyle \int x\cos cx\;dx={\frac {\cos cx}{c^{2}}}+{\frac {x\sin cx}{c}}\,\!}

x

n

cos

c

x

d

x

=

x

n

sin

c

x

c

n

c

x

n

1

sin

c

x

d

x

{\displaystyle \int x^{n}\cos cx\;dx={\frac {x^{n}\sin cx}{c}}-{\frac {n}{c}}\int x^{n-1}\sin cx\;dx\,\!}

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

(

n

=

1

,

3

,

5...

)

{\displaystyle \int _{-{\frac {a}{2}}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(}}n=1,3,5...{\mbox{)}}\,\!}

cos

c

x

x

d

x

=

ln

|

c

x

|

+

i

=

1

(

1

)

i

(

c

x

)

2

i

2

i

(

2

i

)

!

{\displaystyle \int {\frac {\cos cx}{x}}dx=\ln |cx|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(cx)^{2i}}{2i\cdot (2i)!}}\,\!}

cos

c

x

x

n

d

x

=

cos

c

x

(

n

1

)

x

n

1

c

n

1

sin

c

x

x

n

1

d

x

(

n

1

)

{\displaystyle \int {\frac {\cos cx}{x^{n}}}dx=-{\frac {\cos cx}{(n-1)x^{n-1}}}-{\frac {c}{n-1}}\int {\frac {\sin cx}{x^{n-1}}}dx\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}

d

x

cos

c

x

=

1

c

ln

|

tan

(

c

x

2

+

π

4

)

|

{\displaystyle \int {\frac {dx}{\cos cx}}={\frac {1}{c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}

d

x

cos

n

c

x

=

sin

c

x

c

(

n

1

)

c

o

s

n

1

c

x

+

n

2

n

1

d

x

cos

n

2

c

x

(

n

>

1

)

{\displaystyle \int {\frac {dx}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)cos^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{(}}n>1{\mbox{)}}\,\!}

d

x

1

+

cos

c

x

=

1

c

tan

c

x

2

{\displaystyle \int {\frac {dx}{1+\cos cx}}={\frac {1}{c}}\tan {\frac {cx}{2}}\,\!}

d

x

1

cos

c

x

=

1

c

cot

c

x

2

{\displaystyle \int {\frac {dx}{1-\cos cx}}=-{\frac {1}{c}}\cot {\frac {cx}{2}}\,\!}

x

d

x

1

+

cos

c

x

=

x

c

tan

c

x

2

+

2

c

2

ln

|

cos

c

x

2

|

{\displaystyle \int {\frac {x\;dx}{1+\cos cx}}={\frac {x}{c}}\tan {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\cos {\frac {cx}{2}}\right|}

x

d

x

1

cos

c

x

=

x

c

cot

c

x

2

+

2

c

2

ln

|

sin

c

x

2

|

{\displaystyle \int {\frac {x\;dx}{1-\cos cx}}=-{\frac {x}{c}}\cot {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\sin {\frac {cx}{2}}\right|}

cos

c

x

d

x

1

+

cos

c

x

=

x

1

c

tan

c

x

2

{\displaystyle \int {\frac {\cos cx\;dx}{1+\cos cx}}=x-{\frac {1}{c}}\tan {\frac {cx}{2}}\,\!}

cos

c

x

d

x

1

cos

c

x

=

x

1

c

cot

c

x

2

{\displaystyle \int {\frac {\cos cx\;dx}{1-\cos cx}}=-x-{\frac {1}{c}}\cot {\frac {cx}{2}}\,\!}

cos

c

1

x

cos

c

2

x

d

x

=

sin

(

c

1

c

2

)

x

2

(

c

1

c

2

)

+

sin

(

c

1

+

c

2

)

x

2

(

c

1

+

c

2

)

(

|

c

1

|

|

c

2

|

)

{\displaystyle \int \cos c_{1}x\cos c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(}}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}

积分只有tan的函數

编辑

tan

c

x

d

x

=

1

c

ln

|

cos

c

x

|

=

1

c

ln

|

sec

c

x

|

{\displaystyle \int \tan cx\;dx=-{\frac {1}{c}}\ln |\cos cx|\ ={\frac {1}{c}}\ln |\sec cx|\,\!}

tan

n

c

x

d

x

=

1

c

(

n

1

)

tan

n

1

c

x

tan

n

2

c

x

d

x

(for

n

1

)

{\displaystyle \int \tan ^{n}cx\;dx={\frac {1}{c(n-1)}}\tan ^{n-1}cx-\int \tan ^{n-2}cx\;dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

d

x

tan

c

x

+

1

=

x

2

+

1

2

c

ln

|

sin

c

x

+

cos

c

x

|

{\displaystyle \int {\frac {dx}{\tan cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|\,\!}

d

x

tan

c

x

1

=

x

2

+

1

2

c

ln

|

sin

c

x

cos

c

x

|

{\displaystyle \int {\frac {dx}{\tan cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|\,\!}

tan

c

x

d

x

tan

c

x

+

1

=

x

2

1

2

c

ln

|

sin

c

x

+

cos

c

x

|

{\displaystyle \int {\frac {\tan cx\;dx}{\tan cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|\,\!}

tan

c

x

d

x

tan

c

x

1

=

x

2

+

1

2

c

ln

|

sin

c

x

cos

c

x

|

{\displaystyle \int {\frac {\tan cx\;dx}{\tan cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|\,\!}

积分只有sec的函數

编辑

sec

c

x

d

x

=

1

c

ln

|

sec

c

x

+

tan

c

x

|

{\displaystyle \int \sec {cx}\,dx={\frac {1}{c}}\ln {\left|\sec {cx}+\tan {cx}\right|}}

sec

2

x

d

x

=

tan

x

+

C

{\displaystyle \int \sec ^{2}x{\mbox{d}}x=\tan x+C}

sec

n

c

x

d

x

=

sec

n

2

c

x

tan

c

x

c

(

n

1

)

+

n

2

n

1

sec

n

2

c

x

d

x

(for

n

1

)

{\displaystyle \int \sec ^{n}{cx}\,dx={\frac {\sec ^{n-2}{cx}\tan {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{cx}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}

d

x

sec

x

+

1

=

x

tan

x

2

{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}}

积分只有csc的函數

编辑

csc

c

x

d

x

=

1

c

ln

|

tan

(

c

x

2

)

|

{\displaystyle \int \csc {cx}\,dx={\frac {1}{c}}\ln {\left|\tan({\frac {cx}{2}})\right|}}

csc

2

x

d

x

=

cot

x

+

C

{\displaystyle \int \csc ^{2}x{\mbox{d}}x=-\cot x+C}

csc

n

c

x

d

x

=

csc

n

2

c

x

cot

c

x

c

(

n

1

)

+

n

2

n

1

csc

n

2

c

x

d

x

(for

n

1

)

{\displaystyle \int \csc ^{n}{cx}\,dx=-{\frac {\csc ^{n-2}{cx}\cot {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{cx}\,dx\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}\,\!}

积分只有cot的函數

编辑

cot

c

x

d

x

=

1

c

ln

|

sin

c

x

|

{\displaystyle \int \cot cx\;dx={\frac {1}{c}}\ln |\sin cx|\,\!}

cot

n

c

x

d

x

=

1

c

(

n

1

)

cot

n

1

c

x

cot

n

2

c

x

d

x

(for

n

1

)

{\displaystyle \int \cot ^{n}cx\;dx=-{\frac {1}{c(n-1)}}\cot ^{n-1}cx-\int \cot ^{n-2}cx\;dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

d

x

1

+

cot

c

x

=

tan

c

x

d

x

tan

c

x

+

1

{\displaystyle \int {\frac {dx}{1+\cot cx}}=\int {\frac {\tan cx\;dx}{\tan cx+1}}\,\!}

d

x

1

cot

c

x

=

tan

c

x

d

x

tan

c

x

1

{\displaystyle \int {\frac {dx}{1-\cot cx}}=\int {\frac {\tan cx\;dx}{\tan cx-1}}\,\!}

积分只有sin和cos的函數

编辑

d

x

cos

c

x

±

sin

c

x

=

1

c

2

ln

|

tan

(

c

x

2

±

π

8

)

|

{\displaystyle \int {\frac {dx}{\cos cx\pm \sin cx}}={\frac {1}{c{\sqrt {2}}}}\ln \left|\tan \left({\frac {cx}{2}}\pm {\frac {\pi }{8}}\right)\right|}

d

x

(

cos

c

x

±

sin

c

x

)

2

=

1

2

c

tan

(

c

x

π

4

)

{\displaystyle \int {\frac {dx}{(\cos cx\pm \sin cx)^{2}}}={\frac {1}{2c}}\tan \left(cx\mp {\frac {\pi }{4}}\right)}

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 {dx}{(\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 {dx}{(\cos x+\sin x)^{n-2}}}\right]}

cos

c

x

d

x

cos

c

x

+

sin

c

x

=

x

2

+

1

2

c

ln

|

sin

c

x

+

cos

c

x

|

{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}

cos

c

x

d

x

cos

c

x

sin

c

x

=

x

2

1

2

c

ln

|

sin

c

x

cos

c

x

|

{\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}

sin

c

x

d

x

cos

c

x

+

sin

c

x

=

x

2

1

2

c

ln

|

sin

c

x

+

cos

c

x

|

{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}

sin

c

x

d

x

cos

c

x

sin

c

x

=

x

2

1

2

c

ln

|

sin

c

x

cos

c

x

|

{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}

cos

c

x

d

x

sin

c

x

(

1

+

cos

c

x

)

=

1

4

c

tan

2

c

x

2

+

1

2

c

ln

|

tan

c

x

2

|

{\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+\cos cx)}}=-{\frac {1}{4c}}\tan ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|}

cos

c

x

d

x

sin

c

x

(

1

+

cos

c

x

)

=

1

4

c

cot

2

c

x

2

1

2

c

ln

|

tan

c

x

2

|

{\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+-\cos cx)}}=-{\frac {1}{4c}}\cot ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|}

sin

c

x

d

x

cos

c

x

(

1

+

sin

c

x

)

=

1

4

c

cot

2

(

c

x

2

+

π

4

)

+

1

2

c

ln

|

tan

(

c

x

2

+

π

4

)

|

{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1+\sin cx)}}={\frac {1}{4c}}\cot ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}

sin

c

x

d

x

cos

c

x

(

1

sin

c

x

)

=

1

4

c

tan

2

(

c

x

2

+

π

4

)

1

2

c

ln

|

tan

(

c

x

2

+

π

4

)

|

{\displaystyle \int {\frac {\sin cx\;dx}{\cos cx(1-\sin cx)}}={\frac {1}{4c}}\tan ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}

sin

c

x

cos

c

x

d

x

=

1

2

c

sin

2

c

x

{\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}}\sin ^{2}cx\,\!}

sin

c

1

x

cos

c

2

x

d

x

=

cos

(

c

1

+

c

2

)

x

2

(

c

1

+

c

2

)

cos

(

c

1

c

2

)

x

2

(

c

1

c

2

)

(for

|

c

1

|

|

c

2

|

)

{\displaystyle \int \sin c_{1}x\cos c_{2}x\;dx=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}\qquad {\mbox{(for }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}

sin

n

c

x

cos

c

x

d

x

=

1

c

(

n

+

1

)

sin

n

+

1

c

x

(for

n

1

)

{\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

sin

c

x

cos

n

c

x

d

x

=

1

c

(

n

+

1

)

cos

n

+

1

c

x

(for

n

1

)

{\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

sin

n

c

x

cos

m

c

x

d

x

=

sin

n

1

c

x

cos

m

+

1

c

x

c

(

n

+

m

)

+

n

1

n

+

m

sin

n

2

c

x

cos

m

c

x

d

x

(for

m

,

n

>

0

)

{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx=-{\frac {\sin ^{n-1}cx\cos ^{m+1}cx}{c(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}cx\cos ^{m}cx\;dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}

also:

sin

n

c

x

cos

m

c

x

d

x

=

sin

n

+

1

c

x

cos

m

1

c

x

c

(

n

+

m

)

+

m

1

n

+

m

sin

n

c

x

cos

m

2

c

x

d

x

(for

m

,

n

>

0

)

{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;dx={\frac {\sin ^{n+1}cx\cos ^{m-1}cx}{c(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}cx\cos ^{m-2}cx\;dx\qquad {\mbox{(for }}m,n>0{\mbox{)}}\,\!}

d

x

sin

c

x

cos

c

x

=

1

c

ln

|

tan

c

x

|

{\displaystyle \int {\frac {dx}{\sin cx\cos cx}}={\frac {1}{c}}\ln \left|\tan cx\right|}

d

x

sin

c

x

cos

n

c

x

=

1

c

(

n

1

)

cos

n

1

c

x

+

d

x

sin

c

x

cos

n

2

c

x

(for

n

1

)

{\displaystyle \int {\frac {dx}{\sin cx\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}+\int {\frac {dx}{\sin cx\cos ^{n-2}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

d

x

sin

n

c

x

cos

c

x

=

1

c

(

n

1

)

sin

n

1

c

x

+

d

x

sin

n

2

c

x

cos

c

x

(for

n

1

)

{\displaystyle \int {\frac {dx}{\sin ^{n}cx\cos cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}+\int {\frac {dx}{\sin ^{n-2}cx\cos cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

sin

c

x

d

x

cos

n

c

x

=

1

c

(

n

1

)

cos

n

1

c

x

(for

n

1

)

{\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

sin

2

c

x

d

x

cos

c

x

=

1

c

sin

c

x

+

1

c

ln

|

tan

(

π

4

+

c

x

2

)

|

{\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos cx}}=-{\frac {1}{c}}\sin cx+{\frac {1}{c}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {cx}{2}}\right)\right|}

sin

2

c

x

d

x

cos

n

c

x

=

sin

c

x

c

(

n

1

)

cos

n

1

c

x

1

n

1

d

x

cos

n

2

c

x

(for

n

1

)

{\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

sin

n

c

x

d

x

cos

c

x

=

sin

n

1

c

x

c

(

n

1

)

+

sin

n

2

c

x

d

x

cos

c

x

(for

n

1

)

{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos cx}}=-{\frac {\sin ^{n-1}cx}{c(n-1)}}+\int {\frac {\sin ^{n-2}cx\;dx}{\cos cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

sin

n

c

x

d

x

cos

m

c

x

=

sin

n

+

1

c

x

c

(

m

1

)

cos

m

1

c

x

n

m

+

2

m

1

sin

n

c

x

d

x

cos

m

2

c

x

(for

m

1

)

{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n+1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}

also:

sin

n

c

x

d

x

cos

m

c

x

=

sin

n

1

c

x

c

(

n

m

)

cos

m

1

c

x

+

n

1

n

m

sin

n

2

c

x

d

x

cos

m

c

x

(for

m

n

)

{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}=-{\frac {\sin ^{n-1}cx}{c(n-m)\cos ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}cx\;dx}{\cos ^{m}cx}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}

also:

sin

n

c

x

d

x

cos

m

c

x

=

sin

n

1

c

x

c

(

m

1

)

cos

m

1

c

x

n

1

m

1

sin

n

2

c

x

d

x

cos

m

2

c

x

(for

m

1

)

{\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\sin ^{n-1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-2}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}

cos

c

x

d

x

sin

n

c

x

=

1

c

(

n

1

)

sin

n

1

c

x

(for

n

1

)

{\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

cos

2

c

x

d

x

sin

c

x

=

1

c

(

cos

c

x

+

ln

|

tan

c

x

2

|

)

{\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\tan {\frac {cx}{2}}\right|\right)}

cos

2

c

x

d

x

sin

n

c

x

=

1

n

1

(

cos

c

x

c

sin

n

1

c

x

)

+

d

x

sin

n

2

c

x

)

(for

n

1

)

{\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin ^{n}cx}}=-{\frac {1}{n-1}}\left({\frac {\cos cx}{c\sin ^{n-1}cx)}}+\int {\frac {dx}{\sin ^{n-2}cx}}\right)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}

cos

n

c

x

d

x

sin

m

c

x

=

cos

n

+

1

c

x

c

(

m

1

)

sin

m

1

c

x

n

m

+

2

m

1

c

o

s

n

c

x

d

x

sin

m

2

c

x

(for

m

1

)

{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n+1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-m+2}{m-1}}\int {\frac {cos^{n}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}

also:

cos

n

c

x

d

x

sin

m

c

x

=

cos

n

1

c

x

c

(

n

m

)

sin

m

1

c

x

+

n

1

n

m

c

o

s

n

2

c

x

d

x

sin

m

c

x

(for

m

n

)

{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}={\frac {\cos ^{n-1}cx}{c(n-m)\sin ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m}cx}}\qquad {\mbox{(for }}m\neq n{\mbox{)}}\,\!}

also:

cos

n

c

x

d

x

sin

m

c

x

=

cos

n

1

c

x

c

(

m

1

)

sin

m

1

c

x

n

1

m

1

c

o

s

n

2

c

x

d

x

sin

m

2

c

x

(for

m

1

)

{\displaystyle \int {\frac {\cos ^{n}cx\;dx}{\sin ^{m}cx}}=-{\frac {\cos ^{n-1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {cos^{n-2}cx\;dx}{\sin ^{m-2}cx}}\qquad {\mbox{(for }}m\neq 1{\mbox{)}}\,\!}

积分只有sin和tan的函數

编辑

sin

c

x

tan

c

x

d

x

=

1

c

(

ln

|

sec

c

x

+

tan

c

x

|

sin

c

x

)

{\displaystyle \int \sin cx\tan cx\;dx={\frac {1}{c}}(\ln |\sec cx+\tan cx|-\sin cx)\,\!}

tan

n

c

x

d

x

sin

2

c

x

=

1

c

(

n

1

)

tan

n

1

(

c

x

)

(for

n

1

)

{\displaystyle \int {\frac {\tan ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {1}{c(n-1)}}\tan ^{n-1}(cx)\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

积分只有cos和tan的函數

编辑

tan

n

c

x

d

x

cos

2

c

x

=

1

c

(

n

+

1

)

tan

n

+

1

c

x

(for

n

1

)

{\displaystyle \int {\frac {\tan ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\tan ^{n+1}cx\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}

积分只有sin和cot的函數

编辑

cot

n

c

x

d

x

sin

2

c

x

=

1

c

(

n

+

1

)

cot

n

+

1

c

x

(for

n

1

)

{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\sin ^{2}cx}}={\frac {-1}{c(n+1)}}\cot ^{n+1}cx\qquad {\mbox{(for }}n\neq -1{\mbox{)}}\,\!}

积分只有cos和cot的函數

编辑

cot

n

c

x

d

x

cos

2

c

x

=

1

c

(

1

n

)

tan

1

n

c

x

(for

n

1

)

{\displaystyle \int {\frac {\cot ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\tan ^{1-n}cx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}\,\!}

积分只有tan和cot的函數

编辑

tan

m

(

c

x

)

cot

n

(

c

x

)

d

x

=

1

c

(

m

+

n

1

)

tan

m

+

n

1

(

c

x

)

tan

m

2

(

c

x

)

cot

n

(

c

x

)

d

x

(for

m

+

n

1

)

{\displaystyle \int {\frac {\tan ^{m}(cx)}{\cot ^{n}(cx)}}\;dx={\frac {1}{c(m+n-1)}}\tan ^{m+n-1}(cx)-\int {\frac {\tan ^{m-2}(cx)}{\cot ^{n}(cx)}}\;dx\qquad {\mbox{(for }}m+n\neq 1{\mbox{)}}\,\!}

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