integral of x^2arcsin(sqrt(1-x^2))

解答

\int x^{2} arcsin (1 - x^{2}) d x

- \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3}) + C

求解步骤

\int x^{2} arcsin (1 - x^{2}) d x

使用换元积分法

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

提出常数:

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

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

使用分布积分法

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

\int \frac{1}{3} sin^{3} (u) d u = \frac{1}{3} (- cos (u) + \frac{cos ^{3} ( u )}{3})

= - (\frac{1}{3} u sin^{3} (u) - \frac{1}{3} (- cos (u) + \frac{cos ^{3} ( u )}{3}))

u = arcsin (1 - x^{2})

代回

= - (\frac{1}{3} arcsin (1 - x^{2}) sin^{3} (arcsin (1 - x^{2})) - \frac{1}{3} (- cos (arcsin (1 - x^{2})) + \frac{cos ^{3} ( arcsin ( 1 - x ^{2} ) )}{3}))

化简

- (\frac{1}{3} arcsin (1 - x^{2}) sin^{3} (arcsin (1 - x^{2})) - \frac{1}{3} (- cos (arcsin (1 - x^{2})) + \frac{cos ^{3} ( arcsin ( 1 - x ^{2} ) )}{3})) : - \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3})

= - \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3})

解答补常数

= - \frac{1}{3} arcsin (1 - x^{2}) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{9} (- 3 x + x^{3}) + C

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