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

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

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integral of 4/((1+2x)^3)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 u ^{3}} d u

= 4 \cdot \int \frac{1}{2 u ^{3}} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

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

= 4 \cdot \frac{1}{2} (- \frac{1}{2 u ^{2}})

Substitute back

u = 1 + 2 x

= 4 \cdot \frac{1}{2} (- \frac{1}{2 ( 1 + 2 x ) ^{2}})

Simplify

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

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

Add a constant to the solution

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

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