What is the integral of (2-3x)^8 ?

The integral of (2-3x)^8 is 1/27 (-2+3x)^9+C

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integral of (2-3x)^8

\int (2 - 3 x)^{8} d x

\frac{1}{27} (- 2 + 3 x)^{9} + C

Solution steps

\int (2 - 3 x)^{8} d x

Factor

2 - 3 x : - (- 2 + 3 x)

= \int (- (- 2 + 3 x))^{8} d x

Simplify

(- (- 2 + 3 x))^{8} : (- 2 + 3 x)^{8}

= \int (- 2 + 3 x)^{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 = - 2 + 3 x

= \frac{1}{3} \cdot \frac{( - 2 + 3 x ) ^{9}}{9}

Simplify

\frac{1}{3} \cdot \frac{( - 2 + 3 x ) ^{9}}{9} : \frac{1}{27} (- 2 + 3 x)^{9}

= \frac{1}{27} (- 2 + 3 x)^{9}

Add a constant to the solution

= \frac{1}{27} (- 2 + 3 x)^{9} + C