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

The integral of (2x^2-3)^5x is 1/24 (2x^2-3)^6+C

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

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

\frac{1}{24} (2 x^{2} - 3)^{6} + C

Solution steps

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

Apply u-substitution

= \int \frac{u ^{5}}{4} d u

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = 2 x^{2} - 3

= \frac{1}{4} \cdot \frac{( 2 x ^{2} - 3 ) ^{6}}{6}

Simplify

\frac{1}{4} \cdot \frac{( 2 x ^{2} - 3 ) ^{6}}{6} : \frac{1}{24} (2 x^{2} - 3)^{6}

= \frac{1}{24} (2 x^{2} - 3)^{6}

Add a constant to the solution

= \frac{1}{24} (2 x^{2} - 3)^{6} + C

What is the integral of (2x^2-3)^5x ?
The integral of (2x^2-3)^5x is 1/24 (2x^2-3)^6+C