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

The integral of x(2x^2+1)^2 is 1/12 (2x^2+1)^3+C

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integral of x(2x^2+1)^2

\int x (2 x^{2} + 1)^{2} d x

\frac{1}{12} (2 x^{2} + 1)^{3} + C

Solution steps

\int x (2 x^{2} + 1)^{2} d x

Apply u-substitution

: \int \frac{u ^{2}}{4} d u

= \int \frac{u ^{2}}{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^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{1}{4} \cdot \frac{u ^{3}}{3}

Substitute back

u = 2 x^{2} + 1

= \frac{1}{4} \cdot \frac{( 2 x ^{2} + 1 ) ^{3}}{3}

Simplify

\frac{1}{4} \cdot \frac{( 2 x ^{2} + 1 ) ^{3}}{3} : \frac{1}{12} (2 x^{2} + 1)^{3}

= \frac{1}{12} (2 x^{2} + 1)^{3}

Add a constant to the solution

= \frac{1}{12} (2 x^{2} + 1)^{3} + C

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What is the integral of x(2x^2+1)^2 ?
The integral of x(2x^2+1)^2 is 1/12 (2x^2+1)^3+C