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受欢迎的微积分 >

integral of sin^{-5}(x)cos^7(x)

解答

\int sin^{- 5} (x) cos^{7} (x) d x

解答

- \frac{1}{4} csc^{4} (x) + \frac{3}{2} csc^{2} (x) + 3 ln ∣ sin (x) ∣ - \frac{1}{2} sin^{2} (x) + C

求解步骤

\int sin^{- 5} (x) cos^{7} (x) d x

使用三角恒等式改写

= \int (1 - sin^{2} (x))^{3} cos (x) sin^{- 5} (x) d x

使用换元积分法

= \int \frac{( 1 - u ^{2} ) ^{3}}{u ^{5}} d u

乘开

\frac{( 1 - u ^{2} ) ^{3}}{u ^{5}} : \frac{1}{u ^{5}} - \frac{3}{u ^{3}} + \frac{3}{u} - u

= \int \frac{1}{u ^{5}} - \frac{3}{u ^{3}} + \frac{3}{u} - u d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u ^{5}} d u - \int \frac{3}{u ^{3}} d u + \int \frac{3}{u} d u - \int u d u

\int \frac{1}{u ^{5}} d u = - \frac{1}{4 u ^{4}}

\int \frac{3}{u ^{3}} d u = - \frac{3}{2 u ^{2}}

\int \frac{3}{u} d u = 3 ln ∣ u ∣

\int u d u = \frac{u ^{2}}{2}

= - \frac{1}{4 u ^{4}} - (- \frac{3}{2 u ^{2}}) + 3 ln ∣ u ∣ - \frac{u ^{2}}{2}

u = sin (x)

代回

= - \frac{1}{4 sin ^{4} ( x )} - (- \frac{3}{2 sin ^{2} ( x )}) + 3 ln ∣ sin (x) ∣ - \frac{sin ^{2} ( x )}{2}

化简

- \frac{1}{4 sin ^{4} ( x )} - (- \frac{3}{2 sin ^{2} ( x )}) + 3 ln ∣ sin (x) ∣ - \frac{sin ^{2} ( x )}{2} : - \frac{1}{4} csc^{4} (x) + \frac{3}{2} csc^{2} (x) + 3 ln ∣ sin (x) ∣ - \frac{1}{2} sin^{2} (x)

= - \frac{1}{4} csc^{4} (x) + \frac{3}{2} csc^{2} (x) + 3 ln ∣ sin (x) ∣ - \frac{1}{2} sin^{2} (x)

解答补常数

= - \frac{1}{4} csc^{4} (x) + \frac{3}{2} csc^{2} (x) + 3 ln ∣ sin (x) ∣ - \frac{1}{2} sin^{2} (x) + C

作图

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