What is the integral of [x-5/((3x+5)^2)] ?

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

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integral of [x-5/((3x+5)^2)]

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

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

Solution steps

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

\int x d x = \frac{x ^{2}}{2}

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

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

Simplify

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

Add a constant to the solution

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

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