What is the integral of x/((x+2)^7) ?

The integral of x/((x+2)^7) is -1/(5(x+2)^5)+1/(3(x+2)^6)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x/((x+2)^7)

\int \frac{x}{( x + 2 ) ^{7}} d x

- \frac{1}{5 ( x + 2 ) ^{5}} + \frac{1}{3 ( x + 2 ) ^{6}} + C

Solution steps

\int \frac{x}{( x + 2 ) ^{7}} d x

Take the partial fraction of

\frac{x}{( x + 2 ) ^{7}} : \frac{1}{( x + 2 ) ^{6}} - \frac{2}{( x + 2 ) ^{7}}

= \int \frac{1}{( x + 2 ) ^{6}} - \frac{2}{( x + 2 ) ^{7}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{( x + 2 ) ^{6}} d x - \int \frac{2}{( x + 2 ) ^{7}} d x

\int \frac{1}{( x + 2 ) ^{6}} d x = - \frac{1}{5 ( x + 2 ) ^{5}}

\int \frac{2}{( x + 2 ) ^{7}} d x = - \frac{1}{3 ( x + 2 ) ^{6}}

= - \frac{1}{5 ( x + 2 ) ^{5}} - (- \frac{1}{3 ( x + 2 ) ^{6}})

Simplify

= - \frac{1}{5 ( x + 2 ) ^{5}} + \frac{1}{3 ( x + 2 ) ^{6}}

Add a constant to the solution

= - \frac{1}{5 ( x + 2 ) ^{5}} + \frac{1}{3 ( x + 2 ) ^{6}} + C

\int (8 x + 1) cos (\frac{x}{4}) d x

\int \frac{1 0 ^{6 x} + 1}{1 0 ^{3 x}} d x

\int 2 x (x^{2} - 3)^{2} d x

\int 5 t^{\frac{3}{2}} d t

\int \frac{cos ( 2 z )}{cos ( z )} d z

What is the integral of x/((x+2)^7) ?
The integral of x/((x+2)^7) is -1/(5(x+2)^5)+1/(3(x+2)^6)+C