What is the integral of (6x-2)^{-4} ?

The integral of (6x-2)^{-4} is -1/(18(6x-2)^3)+C

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integral of (6x-2)^{-4}

\int (6 x - 2)^{- 4} d x

- \frac{1}{18 ( 6 x - 2 ) ^{3}} + C

Solution steps

\int (6 x - 2)^{- 4} d x

Apply exponent rule:

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

= \int \frac{1}{( 6 x - 2 ) ^{4}} d x

Apply u-substitution

= \int \frac{1}{6 u ^{4}} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u ^{4}} d u

Apply the Power Rule

= \frac{1}{6} (- \frac{1}{3 u ^{3}})

Substitute back

u = 6 x - 2

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

Simplify

\frac{1}{6} (- \frac{1}{3 ( 6 x - 2 ) ^{3}}) : - \frac{1}{18 ( 6 x - 2 ) ^{3}}

= - \frac{1}{18 ( 6 x - 2 ) ^{3}}

Add a constant to the solution

= - \frac{1}{18 ( 6 x - 2 ) ^{3}} + C

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What is the integral of (6x-2)^{-4} ?
The integral of (6x-2)^{-4} is -1/(18(6x-2)^3)+C