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

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

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

integral

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

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

Solution steps

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

Apply u-substitution

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

= \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

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

Apply the Power Rule

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

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

Substitute back

u = 2 x + 1

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

Simplify

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

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

Add a constant to the solution

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

\frac{cos ( \frac{1}{x} )}{x ^{3}}

\frac{1}{1 + cos ^{2} ( x )}

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

x^{3} e^{x}

\frac{2 x ^{2}}{( x ^{2} + 4 )}

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