What is the integral of 1/((1/3 x-8)^5) ?

The integral of 1/((1/3 x-8)^5) is -(243)/(4(x-24)^4)+C

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integral of 1/((1/3 x-8)^5)

\int \frac{1}{( \frac{1}{3} x - 8 ) ^{5}} d x

- \frac{243}{4 ( x - 24 ) ^{4}} + C

Solution steps

\int \frac{1}{( \frac{1}{3} x - 8 ) ^{5}} d x

Take the partial fraction of

\frac{1}{( \frac{1}{3} x - 8 ) ^{5}} : \frac{243}{( x - 24 ) ^{5}}

= \int \frac{243}{( x - 24 ) ^{5}} d x

Take the constant out:

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

= 243 \cdot \int \frac{1}{( x - 24 ) ^{5}} d x

Apply u-substitution

= 243 \cdot \int \frac{1}{u ^{5}} d u

Apply the Power Rule

= 243 (- \frac{1}{4 u ^{4}})

Substitute back

u = x - 24

= 243 (- \frac{1}{4 ( x - 24 ) ^{4}})

Simplify

243 (- \frac{1}{4 ( x - 24 ) ^{4}}) : - \frac{243}{4 ( x - 24 ) ^{4}}

= - \frac{243}{4 ( x - 24 ) ^{4}}

Add a constant to the solution

= - \frac{243}{4 ( x - 24 ) ^{4}} + C

What is the integral of 1/((1/3 x-8)^5) ?
The integral of 1/((1/3 x-8)^5) is -(243)/(4(x-24)^4)+C