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

The integral of x(1-2x^2)^3 is -1/16 (1-2x^2)^4+C

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

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

- \frac{1}{16} (1 - 2 x^{2})^{4} + C

Solution steps

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

Apply u-substitution

: \int - \frac{u ^{3}}{4} d u

= \int - \frac{u ^{3}}{4} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int u^{3} d u

Apply the Power Rule

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

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

Substitute back

u = 1 - 2 x^{2}

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

Simplify

- \frac{1}{4} \cdot \frac{( 1 - 2 x ^{2} ) ^{4}}{4} : - \frac{1}{16} (1 - 2 x^{2})^{4}

= - \frac{1}{16} (1 - 2 x^{2})^{4}

Add a constant to the solution

= - \frac{1}{16} (1 - 2 x^{2})^{4} + C

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What is the integral of x(1-2x^2)^3 ?
The integral of x(1-2x^2)^3 is -1/16 (1-2x^2)^4+C