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

The integral of 1/((3x-5)^8) is -1/(21(3x-5)^7)+C

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

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

- \frac{1}{21 ( 3 x - 5 ) ^{7}} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{3 u ^{8}} d u

= \int \frac{1}{3 u ^{8}} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u ^{8}} d u

Apply the Power Rule

: - \frac{1}{7 u ^{7}}

= \frac{1}{3} (- \frac{1}{7 u ^{7}})

Substitute back

u = 3 x - 5

= \frac{1}{3} (- \frac{1}{7 ( 3 x - 5 ) ^{7}})

Simplify

\frac{1}{3} (- \frac{1}{7 ( 3 x - 5 ) ^{7}}) : - \frac{1}{21 ( 3 x - 5 ) ^{7}}

= - \frac{1}{21 ( 3 x - 5 ) ^{7}}

Add a constant to the solution

= - \frac{1}{21 ( 3 x - 5 ) ^{7}} + C

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What is the integral of 1/((3x-5)^8) ?
The integral of 1/((3x-5)^8) is -1/(21(3x-5)^7)+C