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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用三角换元法

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

使用三角恒等式改写

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

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int 1 - cos (2 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

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

\int 1 d u = u

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

= 2 \cdot \frac{1}{2} (u - \frac{1}{2} sin (2 u))

u = arcsin (\frac{1}{2} x)

代回

= 2 \cdot \frac{1}{2} (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x)))

化简

2 \cdot \frac{1}{2} (arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x))) : arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x))

= arcsin (\frac{1}{2} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{2} x))

解答补常数

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

作图

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derivative f(x)=-16t^2-8t+64

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