What is the integral of (2x^2+5)^2(4x) ?

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

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integral of (2x^2+5)^2(4x)

\int (2 x^{2} + 5)^{2} (4 x) d x

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

Solution steps

\int (2 x^{2} + 5)^{2} \cdot 4 x d x

Take the constant out:

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

= 4 \cdot \int (2 x^{2} + 5)^{2} x d x

Apply u-substitution

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

= 4 \cdot \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

= 4 \cdot \frac{1}{4} \cdot \int u^{2} d u

Apply the Power Rule

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

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

Substitute back

u = 2 x^{2} + 5

= 4 \cdot \frac{1}{4} \cdot \frac{( 2 x ^{2} + 5 ) ^{3}}{3}

Simplify

4 \cdot \frac{1}{4} \cdot \frac{( 2 x ^{2} + 5 ) ^{3}}{3} : \frac{1}{3} (2 x^{2} + 5)^{3}

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

Add a constant to the solution

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

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