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受欢迎的微积分 >

integral of (3x)/(x^2+6x+9)

解答

\int \frac{3 x}{x ^{2} + 6 x + 9} d x

解答

3 (ln ∣ x + 3 ∣ + \frac{3}{x + 3}) + C

求解步骤

\int \frac{3 x}{x ^{2} + 6 x + 9} d x

提出常数:

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

= 3 \cdot \int \frac{x}{x ^{2} + 6 x + 9} d x

对

x^{2} + 6 x + 9

配方:

(x + 3)^{2}

= 3 \cdot \int \frac{x}{( x + 3 ) ^{2}} d x

使用换元积分法

= 3 \cdot \int \frac{u - 3}{u ^{2}} d u

乘开

\frac{u - 3}{u ^{2}} : \frac{1}{u} - \frac{3}{u ^{2}}

= 3 \cdot \int \frac{1}{u} - \frac{3}{u ^{2}} 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 \frac{1}{u} d u - \int \frac{3}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{3}{u ^{2}} d u = - \frac{3}{u}

= 3 (ln ∣ u ∣ - (- \frac{3}{u}))

u = x + 3

代回

= 3 (ln ∣ x + 3 ∣ - (- \frac{3}{x + 3}))

化简

= 3 (ln ∣ x + 3 ∣ + \frac{3}{x + 3})

解答补常数

= 3 (ln ∣ x + 3 ∣ + \frac{3}{x + 3}) + C

作图

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