What is the integral of 24x(2x^2+5)^2 ?

The integral of 24x(2x^2+5)^2 is 2(2x^2+5)^3+C

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integral of 24x(2x^2+5)^2

\int 24 x (2 x^{2} + 5)^{2} d x

2 (2 x^{2} + 5)^{3} + C

Solution steps

\int 24 x (2 x^{2} + 5)^{2} d x

Take the constant out:

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

= 24 \cdot \int x (2 x^{2} + 5)^{2} d x

Apply u-substitution

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

= 24 \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

= 24 \cdot \frac{1}{4} \cdot \int u^{2} d u

Apply the Power Rule

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

= 24 \cdot \frac{1}{4} \cdot \frac{u ^{3}}{3}

Substitute back

u = 2 x^{2} + 5

= 24 \cdot \frac{1}{4} \cdot \frac{( 2 x ^{2} + 5 ) ^{3}}{3}

Simplify

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

= 2 (2 x^{2} + 5)^{3}

Add a constant to the solution

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

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What is the integral of 24x(2x^2+5)^2 ?
The integral of 24x(2x^2+5)^2 is 2(2x^2+5)^3+C