integral of (x+3)/(sqrt(9-x^2))

解答

\int \frac{x + 3}{9 - x ^{2}} d x

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

求解步骤

\int \frac{x + 3}{9 - x ^{2}} d x

使用三角换元法

= \int 3 (sin (u) + 1) d u

提出常数:

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

= 3 \cdot \int sin (u) + 1 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

= 3 (\int sin (u) d u + \int 1 d u)

\int sin (u) d u = - cos (u)

\int 1 d u = u

= 3 (- cos (u) + u)

u = arcsin (\frac{1}{3} x)

代回

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

化简

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

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

解答补常数

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

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