What is the integral of (-30)/((2x+1)^2)+30 ?

The integral of (-30)/((2x+1)^2)+30 is 15(2x+1+1/(2x+1))+C

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integral of (-30)/((2x+1)^2)+30

\int \frac{- 30}{( 2 x + 1 ) ^{2}} + 30 d x

15 (2 x + 1 + \frac{1}{2 x + 1}) + C

Solution steps

\int \frac{- 30}{( 2 x + 1 ) ^{2}} + 30 d x

Apply u-substitution

: \int \frac{15 ( u ^{2} - 1 )}{u ^{2}} d u

= \int \frac{15 ( u ^{2} - 1 )}{u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 15 \cdot \int \frac{u ^{2} - 1}{u ^{2}} d u

Expand

\frac{u ^{2} - 1}{u ^{2}} : 1 - \frac{1}{u ^{2}}

= 15 \cdot \int 1 - \frac{1}{u ^{2}} 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

= 15 (\int 1 d u - \int \frac{1}{u ^{2}} d u)

\int 1 d u = u

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

= 15 (u - (- \frac{1}{u}))

Substitute back

u = 2 x + 1

= 15 (2 x + 1 - (- \frac{1}{2 x + 1}))

Simplify

= 15 (2 x + 1 + \frac{1}{2 x + 1})

Add a constant to the solution

= 15 (2 x + 1 + \frac{1}{2 x + 1}) + C

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What is the integral of (-30)/((2x+1)^2)+30 ?
The integral of (-30)/((2x+1)^2)+30 is 15(2x+1+1/(2x+1))+C