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受欢迎的微积分 >

integral of tan^3(x)sin^2(x)

解答

\int tan^{3} (x) sin^{2} (x) d x

解答

\frac{1}{2} sec^{2} (x) + 2 ln ∣ cos (x) ∣ - \frac{1}{2} cos^{2} (x) + C

求解步骤

\int tan^{3} (x) sin^{2} (x) d x

使用三角恒等式改写

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

使用换元积分法

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

乘开

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

= \int - \frac{1}{u ^{3}} + \frac{2}{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 ^{3}} d u + \int \frac{2}{u} d u - \int u d u

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

\int \frac{2}{u} d u = 2 ln ∣ u ∣

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

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

u = cos (x)

代回

= - (- \frac{1}{2 cos ^{2} ( x )}) + 2 ln ∣ cos (x) ∣ - \frac{cos ^{2} ( x )}{2}

化简

- (- \frac{1}{2 cos ^{2} ( x )}) + 2 ln ∣ cos (x) ∣ - \frac{cos ^{2} ( x )}{2} : \frac{1}{2} sec^{2} (x) + 2 ln ∣ cos (x) ∣ - \frac{1}{2} cos^{2} (x)

= \frac{1}{2} sec^{2} (x) + 2 ln ∣ cos (x) ∣ - \frac{1}{2} cos^{2} (x)

解答补常数

= \frac{1}{2} sec^{2} (x) + 2 ln ∣ cos (x) ∣ - \frac{1}{2} cos^{2} (x) + C

作图

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