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受欢迎的微积分 >

integral of (x-8)/((x+4)^2+64)

解答

\int \frac{x - 8}{( x + 4 ) ^{2} + 64} d x

解答

\frac{1}{2} ln x^{2} + 8 x + 80 - \frac{3}{2} arctan (\frac{1}{8} (x + 4)) + C

求解步骤

\int \frac{x - 8}{( x + 4 ) ^{2} + 64} d x

使用换元积分法

= \int \frac{u - 12}{u ^{2} + 64} d u

乘开

\frac{u - 12}{u ^{2} + 64} : \frac{u}{u ^{2} + 64} - \frac{12}{u ^{2} + 64}

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 \frac{u}{u ^{2} + 64} d u - \int \frac{12}{u ^{2} + 64} d u

\int \frac{u}{u ^{2} + 64} d u = \frac{1}{2} ln u^{2} + 64

\int \frac{12}{u ^{2} + 64} d u = \frac{3}{2} arctan (\frac{u}{8})

= \frac{1}{2} ln u^{2} + 64 - \frac{3}{2} arctan (\frac{u}{8})

u = x + 4

代回

= \frac{1}{2} ln (x + 4)^{2} + 64 - \frac{3}{2} arctan (\frac{x + 4}{8})

化简

\frac{1}{2} ln (x + 4)^{2} + 64 - \frac{3}{2} arctan (\frac{x + 4}{8}) : \frac{1}{2} ln x^{2} + 8 x + 80 - \frac{3}{2} arctan (\frac{1}{8} (x + 4))

= \frac{1}{2} ln x^{2} + 8 x + 80 - \frac{3}{2} arctan (\frac{1}{8} (x + 4))

解答补常数

= \frac{1}{2} ln x^{2} + 8 x + 80 - \frac{3}{2} arctan (\frac{1}{8} (x + 4)) + C

作图

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