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受欢迎的微积分 >

integral of (sqrt(1-x^2))/(x^4)

解答

\int \frac{1 - x ^{2}}{x ^{4}} d x

解答

- \frac{( 1 - x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}} + C

求解步骤

\int \frac{1 - x ^{2}}{x ^{4}} d x

使用三角换元法

= \int \frac{cos ^{2} ( u )}{sin ^{4} ( u )} d u

使用三角恒等式改写

= \int \frac{cot ^{2} ( u )}{sin ^{2} ( u )} d u

使用换元积分法

= \int - v^{2} d v

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int v^{2} d v

使用幂法则

= - \frac{v ^{3}}{3}

代回

= - \frac{cot ^{3} ( arcsin ( x ) )}{3}

化简

- \frac{cot ^{3} ( arcsin ( x ) )}{3} : - \frac{( 1 - x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}}

= - \frac{( 1 - x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}}

解答补常数

= - \frac{( 1 - x ^{2} ) ^{\frac{3}{2}}}{3 x ^{3}} + C

作图

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