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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u ^{2}}{3} d u

= 3 \cdot \int \frac{u ^{2}}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int u^{2} d u

Apply the Power Rule

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

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

Substitute back

u = x^{3} + 5

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

Simplify

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

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

Add a constant to the solution

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

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