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

The integral of (3x-5)^8 is 1/27 (3x-5)^9+C

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integral of (3x-5)^8

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

\frac{1}{27} (3 x - 5)^{9} + C

Solution steps

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

Apply u-substitution

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

Apply the Power Rule

= \frac{1}{3} \cdot \frac{u ^{9}}{9}

Substitute back

u = 3 x - 5

= \frac{1}{3} \cdot \frac{( 3 x - 5 ) ^{9}}{9}

Simplify

\frac{1}{3} \cdot \frac{( 3 x - 5 ) ^{9}}{9} : \frac{1}{27} (3 x - 5)^{9}

= \frac{1}{27} (3 x - 5)^{9}

Add a constant to the solution

= \frac{1}{27} (3 x - 5)^{9} + C