What is the integral of x(5x^2-1)^3 ?

The integral of x(5x^2-1)^3 is 1/40 (5x^2-1)^4+C

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integral of x(5x^2-1)^3

\int x (5 x^{2} - 1)^{3} d x

\frac{1}{40} (5 x^{2} - 1)^{4} + C

Solution steps

\int x (5 x^{2} - 1)^{3} d x

Apply u-substitution

: \int \frac{u ^{3}}{10} d u

= \int \frac{u ^{3}}{10} d u

Take the constant out:

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

= \frac{1}{10} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{1}{10} \cdot \frac{u ^{4}}{4}

Substitute back

u = 5 x^{2} - 1

= \frac{1}{10} \cdot \frac{( 5 x ^{2} - 1 ) ^{4}}{4}

Simplify

\frac{1}{10} \cdot \frac{( 5 x ^{2} - 1 ) ^{4}}{4} : \frac{1}{40} (5 x^{2} - 1)^{4}

= \frac{1}{40} (5 x^{2} - 1)^{4}

Add a constant to the solution

= \frac{1}{40} (5 x^{2} - 1)^{4} + C

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What is the integral of x(5x^2-1)^3 ?
The integral of x(5x^2-1)^3 is 1/40 (5x^2-1)^4+C