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受欢迎的微积分 >

integral of (4x)/(sqrt(-x^2-8x-7))

解答

\int \frac{4 x}{- x ^{2} - 8 x - 7} d x

解答

4 (- - x^{2} - 8 x - 7 - 4 arcsin (\frac{1}{3} (x + 4))) + C

求解步骤

\int \frac{4 x}{- x ^{2} - 8 x - 7} d x

提出常数:

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

= 4 \cdot \int \frac{x}{- x ^{2} - 8 x - 7} d x

对

- x^{2} - 8 x - 7

配方:

- (x + 4)^{2} + 9

= 4 \cdot \int \frac{x}{- ( x + 4 ) ^{2} + 9} d x

使用换元积分法

= 4 \cdot \int \frac{u - 4}{- u ^{2} + 9} d u

乘开

\frac{u - 4}{- u ^{2} + 9} : \frac{u}{- u ^{2} + 9} - \frac{4}{- u ^{2} + 9}

= 4 \cdot \int \frac{u}{- u ^{2} + 9} - \frac{4}{- u ^{2} + 9} 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

= 4 (\int \frac{u}{- u ^{2} + 9} d u - \int \frac{4}{- u ^{2} + 9} d u)

\int \frac{u}{- u ^{2} + 9} d u = - - u^{2} + 9

\int \frac{4}{- u ^{2} + 9} d u = 4 arcsin (\frac{1}{3} u)

= 4 (- - u^{2} + 9 - 4 arcsin (\frac{1}{3} u))

u = x + 4

代回

= 4 (- - (x + 4)^{2} + 9 - 4 arcsin (\frac{1}{3} (x + 4)))

乘开

- (x + 4)^{2} + 9 : - x^{2} - 8 x - 7

= 4 (- - x^{2} - 8 x - 7 - 4 arcsin (\frac{1}{3} (x + 4)))

解答补常数

= 4 (- - x^{2} - 8 x - 7 - 4 arcsin (\frac{1}{3} (x + 4))) + C

作图

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