What is the integral of 6(2x-3)^{-6} ?

The integral of 6(2x-3)^{-6} is -3/(5(2x-3)^5)+C

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integral of 6(2x-3)^{-6}

\int 6 (2 x - 3)^{- 6} d x

- \frac{3}{5 ( 2 x - 3 ) ^{5}} + C

Solution steps

\int 6 (2 x - 3)^{- 6} d x

Take the constant out:

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

= 6 \cdot \int (2 x - 3)^{- 6} d x

Apply u-substitution

: \int \frac{1}{2 u ^{6}} d u

= 6 \cdot \int \frac{1}{2 u ^{6}} d u

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{6}} d u

Apply the Power Rule

: - \frac{1}{5 u ^{5}}

= 6 \cdot \frac{1}{2} (- \frac{1}{5 u ^{5}})

Substitute back

u = 2 x - 3

= 6 \cdot \frac{1}{2} (- \frac{1}{5 ( 2 x - 3 ) ^{5}})

Simplify

6 \cdot \frac{1}{2} (- \frac{1}{5 ( 2 x - 3 ) ^{5}}) : - \frac{3}{5 ( 2 x - 3 ) ^{5}}

= - \frac{3}{5 ( 2 x - 3 ) ^{5}}

Add a constant to the solution

= - \frac{3}{5 ( 2 x - 3 ) ^{5}} + C

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What is the integral of 6(2x-3)^{-6} ?
The integral of 6(2x-3)^{-6} is -3/(5(2x-3)^5)+C