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受欢迎的微积分 >

integral of xsqrt(2-\sqrt{1-x^2)}

解答

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

解答

- \frac{2}{5} (2 - 1 - x^{2})^{\frac{5}{2}} + \frac{4}{3} (2 - 1 - x^{2})^{\frac{3}{2}} + C

求解步骤

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

使用换元积分法

= \int - u 2 - u d u

提出常数:

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

= - \int u 2 - u d u

使用换元积分法

= - \int - (- v + 2) v d v

提出常数:

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

= - (- \int (- v + 2) v d v)

乘开

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

= - (- \int - v^{\frac{3}{2}} + 2 v d v)

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 v^{\frac{3}{2}} d v + \int 2 v d v))

\int v^{\frac{3}{2}} d v = \frac{2}{5} v^{\frac{5}{2}}

\int 2 v d v = \frac{4}{3} v^{\frac{3}{2}}

= - (- (- \frac{2}{5} v^{\frac{5}{2}} + \frac{4}{3} v^{\frac{3}{2}}))

代回

= - (- (- \frac{2}{5} (2 - 1 - x^{2})^{\frac{5}{2}} + \frac{4}{3} (2 - 1 - x^{2})^{\frac{3}{2}}))

化简

= - \frac{2}{5} (2 - 1 - x^{2})^{\frac{5}{2}} + \frac{4}{3} (2 - 1 - x^{2})^{\frac{3}{2}}

解答补常数

= - \frac{2}{5} (2 - 1 - x^{2})^{\frac{5}{2}} + \frac{4}{3} (2 - 1 - x^{2})^{\frac{3}{2}} + C

作图

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\int (arccos (x))^{2} d x

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