What is the integral of (2x^2+3)^2*5x ?

The integral of (2x^2+3)^2*5x is 5/12 (2x^2+3)^3+C

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integral of (2x^2+3)^2*5x

\int (2 x^{2} + 3)^{2} \cdot 5 x d x

\frac{5}{12} (2 x^{2} + 3)^{3} + C

Solution steps

\int (2 x^{2} + 3)^{2} \cdot 5 x d x

Take the constant out:

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

= 5 \cdot \int (2 x^{2} + 3)^{2} x d x

Apply u-substitution

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

= 5 \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

= 5 \cdot \frac{1}{4} \cdot \int u^{2} d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{4} \cdot \frac{u ^{3}}{3}

Substitute back

u = 2 x^{2} + 3

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

Simplify

5 \cdot \frac{1}{4} \cdot \frac{( 2 x ^{2} + 3 ) ^{3}}{3} : \frac{5}{12} (2 x^{2} + 3)^{3}

= \frac{5}{12} (2 x^{2} + 3)^{3}

Add a constant to the solution

= \frac{5}{12} (2 x^{2} + 3)^{3} + C

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What is the integral of (2x^2+3)^2*5x ?
The integral of (2x^2+3)^2*5x is 5/12 (2x^2+3)^3+C