What is the integral of x/((x+2)^4) ?

The integral of x/((x+2)^4) is -1/(2(x+2)^2)+2/(3(x+2)^3)+C

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

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

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

Solution steps

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

Take the partial fraction of

\frac{x}{( x + 2 ) ^{4}} : \frac{1}{( x + 2 ) ^{3}} - \frac{2}{( x + 2 ) ^{4}}

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

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

\int \frac{2}{( x + 2 ) ^{4}} d x = - \frac{2}{3 ( x + 2 ) ^{3}}

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

Simplify

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

Add a constant to the solution

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

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What is the integral of x/((x+2)^4) ?
The integral of x/((x+2)^4) is -1/(2(x+2)^2)+2/(3(x+2)^3)+C