What is the integral of 90(2x-5)^8 ?

The integral of 90(2x-5)^8 is 5(2x-5)^9+C

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integral of 90(2x-5)^8

\int 90 (2 x - 5)^{8} d x

5 (2 x - 5)^{9} + C

Solution steps

\int 90 (2 x - 5)^{8} d x

Take the constant out:

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

= 90 \cdot \int (2 x - 5)^{8} d x

Apply u-substitution

: \int \frac{u ^{8}}{2} d u

= 90 \cdot \int \frac{u ^{8}}{2} d u

Take the constant out:

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

= 90 \cdot \frac{1}{2} \cdot \int u^{8} d u

Apply the Power Rule

: \frac{u ^{9}}{9}

= 90 \cdot \frac{1}{2} \cdot \frac{u ^{9}}{9}

Substitute back

u = 2 x - 5

= 90 \cdot \frac{1}{2} \cdot \frac{( 2 x - 5 ) ^{9}}{9}

Simplify

90 \cdot \frac{1}{2} \cdot \frac{( 2 x - 5 ) ^{9}}{9} : 5 (2 x - 5)^{9}

= 5 (2 x - 5)^{9}

Add a constant to the solution

= 5 (2 x - 5)^{9} + C

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