What is the integral of 5x(2+3x^2)^8 ?

The integral of 5x(2+3x^2)^8 is 5/54 (2+3x^2)^9+C

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integral of 5x(2+3x^2)^8

\int 5 x (2 + 3 x^{2})^{8} d x

\frac{5}{54} (2 + 3 x^{2})^{9} + C

Solution steps

\int 5 x (2 + 3 x^{2})^{8} d x

Take the constant out:

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

= 5 \cdot \int x (2 + 3 x^{2})^{8} d x

Apply u-substitution

: \int \frac{u ^{8}}{6} d u

= 5 \cdot \int \frac{u ^{8}}{6} d u

Take the constant out:

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

= 5 \cdot \frac{1}{6} \cdot \int u^{8} d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{6} \cdot \frac{u ^{9}}{9}

Substitute back

u = 2 + 3 x^{2}

= 5 \cdot \frac{1}{6} \cdot \frac{( 2 + 3 x ^{2} ) ^{9}}{9}

Simplify

5 \cdot \frac{1}{6} \cdot \frac{( 2 + 3 x ^{2} ) ^{9}}{9} : \frac{5}{54} (2 + 3 x^{2})^{9}

= \frac{5}{54} (2 + 3 x^{2})^{9}

Add a constant to the solution

= \frac{5}{54} (2 + 3 x^{2})^{9} + C

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What is the integral of 5x(2+3x^2)^8 ?
The integral of 5x(2+3x^2)^8 is 5/54 (2+3x^2)^9+C