What is the integral of (2x+3)^{-4} ?

The integral of (2x+3)^{-4} is -1/(6(2x+3)^3)+C

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integral of (2x+3)^{-4}

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

- \frac{1}{6 ( 2 x + 3 ) ^{3}} + C

Solution steps

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

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

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

Apply u-substitution

: \int \frac{1}{2 u ^{4}} d u

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

Apply the Power Rule

: - \frac{1}{3 u ^{3}}

= \frac{1}{2} (- \frac{1}{3 u ^{3}})

Substitute back

u = 2 x + 3

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

Simplify

\frac{1}{2} (- \frac{1}{3 ( 2 x + 3 ) ^{3}}) : - \frac{1}{6 ( 2 x + 3 ) ^{3}}

= - \frac{1}{6 ( 2 x + 3 ) ^{3}}

Add a constant to the solution

= - \frac{1}{6 ( 2 x + 3 ) ^{3}} + C

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What is the integral of (2x+3)^{-4} ?
The integral of (2x+3)^{-4} is -1/(6(2x+3)^3)+C