What is the integral of x^2(2x^3-3)^4 ?

The integral of x^2(2x^3-3)^4 is 1/30 (2x^3-3)^5+C

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integral of x^2(2x^3-3)^4

\int x^{2} (2 x^{3} - 3)^{4} d x

\frac{1}{30} (2 x^{3} - 3)^{5} + C

Solution steps

\int x^{2} (2 x^{3} - 3)^{4} d x

Apply u-substitution

= \int \frac{u ^{4}}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int u^{4} d u

Apply the Power Rule

= \frac{1}{6} \cdot \frac{u ^{5}}{5}

Substitute back

u = 2 x^{3} - 3

= \frac{1}{6} \cdot \frac{( 2 x ^{3} - 3 ) ^{5}}{5}

Simplify

\frac{1}{6} \cdot \frac{( 2 x ^{3} - 3 ) ^{5}}{5} : \frac{1}{30} (2 x^{3} - 3)^{5}

= \frac{1}{30} (2 x^{3} - 3)^{5}

Add a constant to the solution

= \frac{1}{30} (2 x^{3} - 3)^{5} + C

What is the integral of x^2(2x^3-3)^4 ?
The integral of x^2(2x^3-3)^4 is 1/30 (2x^3-3)^5+C