What is the integral of (2x+1)/((x+3)^3) ?

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

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integral of (2x+1)/((x+3)^3)

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

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

Solution steps

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

Apply u-substitution

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

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

Expand

\frac{2 u - 5}{u ^{3}} : \frac{2}{u ^{2}} - \frac{5}{u ^{3}}

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

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

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

= - \frac{2}{u} - (- \frac{5}{2 u ^{2}})

Substitute back

u = x + 3

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

Simplify

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

Add a constant to the solution

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

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