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受欢迎的微积分 >

integral of (sin^3(θ))/(cos^6(θ))

解答

\int \frac{sin ^{3} ( θ )}{cos ^{6} ( θ )} d θ

解答

\frac{1}{5} sec^{5} (θ) - \frac{1}{3} sec^{3} (θ) + C

求解步骤

\int \frac{sin ^{3} ( θ )}{cos ^{6} ( θ )} d θ

使用三角恒等式改写

= \int \frac{( 1 - cos ^{2} ( θ ) ) sin ( θ )}{cos ^{6} ( θ )} d θ

使用换元积分法

= \int - \frac{1 - u ^{2}}{u ^{6}} d u

乘开

- \frac{1 - u ^{2}}{u ^{6}} : - \frac{1}{u ^{6}} + \frac{1}{u ^{4}}

= \int - \frac{1}{u ^{6}} + \frac{1}{u ^{4}} 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 ^{6}} d u + \int \frac{1}{u ^{4}} d u

\int \frac{1}{u ^{6}} d u = - \frac{1}{5 u ^{5}}

\int \frac{1}{u ^{4}} d u = - \frac{1}{3 u ^{3}}

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

u = cos (θ)

代回

= - (- \frac{1}{5 cos ^{5} ( θ )}) - \frac{1}{3 cos ^{3} ( θ )}

化简

- (- \frac{1}{5 cos ^{5} ( θ )}) - \frac{1}{3 cos ^{3} ( θ )} : \frac{1}{5} sec^{5} (θ) - \frac{1}{3} sec^{3} (θ)

= \frac{1}{5} sec^{5} (θ) - \frac{1}{3} sec^{3} (θ)

解答补常数

= \frac{1}{5} sec^{5} (θ) - \frac{1}{3} sec^{3} (θ) + C

作图

流行的例子

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\int \frac{sin ( 2 x )}{2} d x

(\partial)/(\partial x)(x^zy^{1-z})

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derivative-3e^{x+4}+e^3

d er i v a t i v e - 3 e^{x + 4} + e^{3}

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\frac{d}{d x} (\frac{- x ^{3}}{y ^{3}})