zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (3x)/((x+5)^2)

解答

\int \frac{3 x}{( x + 5 ) ^{2}} d x

解答

3 (ln ∣ x + 5 ∣ + \frac{5}{x + 5}) + C

求解步骤

\int \frac{3 x}{( x + 5 ) ^{2}} 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 + 5 ) ^{2}} d x

使用换元积分法

= 3 \cdot \int \frac{u - 5}{u ^{2}} d u

乘开

\frac{u - 5}{u ^{2}} : \frac{1}{u} - \frac{5}{u ^{2}}

= 3 \cdot \int \frac{1}{u} - \frac{5}{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{5}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{5}{u ^{2}} d u = - \frac{5}{u}

= 3 (ln ∣ u ∣ - (- \frac{5}{u}))

u = x + 5

代回

= 3 (ln ∣ x + 5 ∣ - (- \frac{5}{x + 5}))

化简

= 3 (ln ∣ x + 5 ∣ + \frac{5}{x + 5})

解答补常数

= 3 (ln ∣ x + 5 ∣ + \frac{5}{x + 5}) + C

作图

流行的例子

integral of 1/(6x^2-5x-6)

\int \frac{1}{6 x ^{2} - 5 x - 6} d x

integral of 2/(4x^2+3)

\int \frac{2}{4 x ^{2} + 3} d x

integral of t^2-3t+2

\int t^{2} - 3 t + 2 d t

integral of 2x*x^2

\int 2 x \cdot x^{2} d x

integral of x^3e^{-x/c}

\int x^{3} e^{- \frac{x}{c}} d x