What is the integral of (2x+1)^{-5} ?

The integral of (2x+1)^{-5} is -1/(8(2x+1)^4)+C

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integral of (2x+1)^{-5}

\int (2 x + 1)^{- 5} d x

- \frac{1}{8 ( 2 x + 1 ) ^{4}} + C

Solution steps

\int (2 x + 1)^{- 5} d x

Apply u-substitution

: \int \frac{1}{2 u ^{5}} d u

= \int \frac{1}{2 u ^{5}} 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 ^{5}} d u

Apply the Power Rule

: - \frac{1}{4 u ^{4}}

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

Substitute back

u = 2 x + 1

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

Simplify

\frac{1}{2} (- \frac{1}{4 ( 2 x + 1 ) ^{4}}) : - \frac{1}{8 ( 2 x + 1 ) ^{4}}

= - \frac{1}{8 ( 2 x + 1 ) ^{4}}

Add a constant to the solution

= - \frac{1}{8 ( 2 x + 1 ) ^{4}} + C

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What is the integral of (2x+1)^{-5} ?
The integral of (2x+1)^{-5} is -1/(8(2x+1)^4)+C