(十一) 含有三角函数函数的积分

$$ 83.\,\int\!\! \sin x dx = -\cos x +C $$

$$ 84.\,\int\!\! \cos x dx= \sin x +C $$

$$ 85.\,\int\!\! \tan x dx -\ln \vert \cos x \vert +C $$

$$ 86.\,\int\!\! ctgx dx =\ln \vert \sin x \vert +C $$

$$ 87.\,\int\!\! \sec x dx = \ln \vert \tan (\frac{ \pi}{4} + \frac{x}{2}) \vert + C = \ln \vert \sec x + \tan x \vert +C $$

$$ 88.\,\int\!\! \csc x dx= \ln \vert \tan \frac{x}{2} \vert +C = \ln \vert \csc x - ctg x \vert +C $$

$$ 89.\,\int\!\! \sec^2 x dx = \tan x +C $$

$$ 90.\,\int\!\! \csc ^2 x dx = - ctgx +C $$

$$ 91.\,\int\!\! \sec x \tan x dx = \sec x +C $$

$$ 92.\,\int\!\! \csc x dx ctgx dx = -\csc x +C $$

$$ 93.\,\int\!\! \sin ^2 x dx = \frac{x}{2}- \frac{1}{4}\sin 2x +C $$

$$ 94.\,\int\!\! \cos ^2 x dx = \frac{x}{2} + \frac{1}{4}\sin 2x +C $$

$$ 95.\,\int\!\! \sin ^n x dx = - \frac{1}{n}\sin ^{n-1}x \cos x + \frac{n-1}{n} \int\!\! \sin ^{n-2}dx $$

$$ 96.\,\int\!\! \cos ^ n x dx = \frac{1}{n}\cos^{ n-1}x \sin x + \frac{n-1}{n} \int\!\! \cos^{n-2} x dx $$

$$ 97.\,\int\!\! \frac{dx}{\sin ^ n x} = - \frac{1}{n-1} . \frac{\cos x}{\sin ^{n-1}x}+\frac{n-2}{n-1} \int\!\! \frac{dx}{\sin^{n-2}x} $$

$$ 98.\,\int\!\! \frac{dx}{\cos ^n x}= \frac{1}{n-1}.\frac{\sin x}{\cos ^{n-1}x}+\frac{n-2}{n-1}\int\!\! \frac{dx}{\cos x^{n-2}x} $$

$$ 99.\,\int\!\! \cos ^ m \sin ^n x dx =\frac{1}{m+n}\cos^{m-1}x \sin ^{n+1}x + \frac{m-1}{m+n} \int\!\! \cos ^ {m-2} x \sin ^n x dx $$

$$ \qquad = -\frac{1}{m+1}\cos ^{m+1}x \sin ^{n-1}x + \frac{n-1}{m+n} \int\!\! \cos ^m x \sin ^{n-2} x dx $$

$$ 100.\,\int\!\! \sin ax \cos bx dx = - \frac{1}{2(a+b)}\cos (a+b)x - \frac{1}{2(a-b)} \cos (a-b)x +C $$

$$ 101.\,\int\!\! \sin ax \sin bx dx = - \frac{1}{2(a+b)} \sin (a+b) x + \frac{1}{2(a-b)} \sin (a-b)x +C $$

$$ 102.\,\int\!\! \cos ax \cos bx dx =\frac{1}{2(a+b)} \sin (a+b)x + \frac{1}{2(a-b)} \sin (a-b)x +C $$

$$ 103.\,\int\!\! \frac{dx}{a+b\sin x} = \frac{2}{\sqrt{a^2-b^2}}\arctan \frac{\arctan \frac{x}{2}+b}{\sqrt{a^2-b^2}} +C \qquad ( a^2 > b^2 ) $$

$$ 104.\,\int\!\! \frac{dx}{a+b \sin x} = \frac{1}{\sqrt{b^2-a^2}} \ln \Bigg \vert \frac{\arctan \frac{x}{2}+b - \sqrt{b^2-a^2}}{ \arctan \frac {x}{2}+b+ \sqrt {b^2-a^2}} \Bigg \vert +C \qquad (a^2 < b^2) $$

$$ 105.\,\int\!\! \frac{dx}{a+b \cos x} = \frac{2}{a+b} \sqrt{ \frac{a+b}{a-b}} \arctan \Bigg ( \sqrt { \frac{a-b}{a+b}} \tan \frac{x}{2} \Bigg ) +C \qquad (a^2>b^2) $$

$$ 106.\,\int\!\! \frac{dx}{a+b \cos x}= \frac{1}{a+b}\sqrt{ \frac{a+b}{b-a}} \ln \Bigg \vert \frac{\tan \frac{x}{2}+ \sqrt {\frac{a+b}{b-a}}}{\tan \frac{x}{2}- \sqrt{\frac{a+b}{b-a}}} \Bigg \vert +C \qquad (a^2 < b^2) $$

$$ 107.\,\int\!\! \frac{dx}{a^2\cos ^2x + b^2 \sin ^2 x}= \frac{1}{ab} \arctan (\frac{b}{a}\tan x ) +C $$

$$ 108.\,\int\!\! \frac{dx}{a^2 \cos ^2x -b^2 \sin ^2 x} = \frac{1}{2ab} \ln \Big \vert \frac{b \tan x +a }{b \tan x -a} \Big \vert +C $$

$$ 109.\,\int\!\! x \sin ax dx = \frac{1}{a^2} \sin ax - \frac{1}{a} x \cos ax +C $$

$$ 110.\,\int\!\! x^2 \sin ax dx = -\frac{1}{a}x^2 \cos ax + \frac{2}{a^2}x\sin ax + \frac{2}{a^3}\cos ax +C $$

$$ 111.\,\int\!\! x \cos ax dx = \frac{1}{a^2} \cos ax + \frac{1}{a}x \sin ax +C $$

$$ 112.\,\int\!\! x^2 \cos ax dx = \frac{1}{a} x^2 \sin ax + \frac{2}{a^2}x \cos ax - \frac{2}{a^3}\sin ax +C $$

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