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受欢迎的微积分 >

integral of (arcsin(x))/((1-x^2)^{3/2)}

解答

\int \frac{arcsin ( x )}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x

解答

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

求解步骤

\int \frac{arcsin ( x )}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x

使用换元积分法

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

使用三角恒等式改写

= \int \frac{2 u}{1 + cos ( 2 u )} d u

提出常数:

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

= 2 \cdot \int \frac{u}{1 + cos ( 2 u )} d u

使用分布积分法

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

\int \frac{1}{2} tan (u) d u = - \frac{1}{2} ln ∣ cos (u) ∣

= 2 (\frac{1}{2} u tan (u) - (- \frac{1}{2} ln ∣ cos (u) ∣))

u = arcsin (x)

代回

= 2 (\frac{1}{2} arcsin (x) tan (arcsin (x)) - (- \frac{1}{2} ln ∣ cos (arcsin (x)) ∣))

化简

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

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

解答补常数

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

作图

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