What is the integral of x/((x+3)^5) ?

The integral of x/((x+3)^5) is -1/(3(x+3)^3)+3/(4(x+3)^4)+C

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

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

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

Solution steps

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

Apply u-substitution

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

Expand

\frac{u - 3}{u ^{5}} : \frac{1}{u ^{4}} - \frac{3}{u ^{5}}

= \int \frac{1}{u ^{4}} - \frac{3}{u ^{5}} d u

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}{u ^{4}} d u - \int \frac{3}{u ^{5}} d u

\int \frac{1}{u ^{4}} d u = - \frac{1}{3 u ^{3}}

\int \frac{3}{u ^{5}} d u = - \frac{3}{4 u ^{4}}

= - \frac{1}{3 u ^{3}} - (- \frac{3}{4 u ^{4}})

Substitute back

u = x + 3

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

Simplify

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

Add a constant to the solution

= - \frac{1}{3 ( x + 3 ) ^{3}} + \frac{3}{4 ( x + 3 ) ^{4}} + C

What is the integral of x/((x+3)^5) ?
The integral of x/((x+3)^5) is -1/(3(x+3)^3)+3/(4(x+3)^4)+C