What is the integral of x(3x^2-5)^7 ?

The integral of x(3x^2-5)^7 is 1/48 (3x^2-5)^8+C

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integral of x(3x^2-5)^7

\int x (3 x^{2} - 5)^{7} d x

\frac{1}{48} (3 x^{2} - 5)^{8} + C

Solution steps

\int x (3 x^{2} - 5)^{7} d x

Apply u-substitution

: \int \frac{u ^{7}}{6} d u

= \int \frac{u ^{7}}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= \frac{1}{6} \cdot \frac{u ^{8}}{8}

Substitute back

u = 3 x^{2} - 5

= \frac{1}{6} \cdot \frac{( 3 x ^{2} - 5 ) ^{8}}{8}

Simplify

\frac{1}{6} \cdot \frac{( 3 x ^{2} - 5 ) ^{8}}{8} : \frac{1}{48} (3 x^{2} - 5)^{8}

= \frac{1}{48} (3 x^{2} - 5)^{8}

Add a constant to the solution

= \frac{1}{48} (3 x^{2} - 5)^{8} + C

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What is the integral of x(3x^2-5)^7 ?
The integral of x(3x^2-5)^7 is 1/48 (3x^2-5)^8+C