What is the integral of (3x^2-5)^4x ?

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

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integral of (3x^2-5)^4x

\int (3 x^{2} - 5)^{4} x d x

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

Solution steps

\int (3 x^{2} - 5)^{4} x d x

Apply u-substitution

: \int \frac{( 3 u - 5 ) ^{4}}{2} d u

= \int \frac{( 3 u - 5 ) ^{4}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int (3 u - 5)^{4} d u

Apply u-substitution

: \int \frac{v ^{4}}{3} d v

= \frac{1}{2} \cdot \int \frac{v ^{4}}{3} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{3} \cdot \int v^{4} d v

Apply the Power Rule

: \frac{v ^{5}}{5}

= \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{v ^{5}}{5}

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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