What is the integral of (3x^2+5x)^4(6x+5) ?

The integral of (3x^2+5x)^4(6x+5) is 1/1244160 (36x^2+60x)^5+C

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integral of (3x^2+5x)^4(6x+5)

\int (3 x^{2} + 5 x)^{4} (6 x + 5) d x

\frac{1}{1244160} (36 x^{2} + 60 x)^{5} + C

Solution steps

\int (3 x^{2} + 5 x)^{4} (6 x + 5) d x

Apply u-substitution

: \int \frac{u ( u ^{2} - 25 ) ^{4}}{124416} d u

= \int \frac{u ( u ^{2} - 25 ) ^{4}}{124416} d u

Take the constant out:

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

= \frac{1}{124416} \cdot \int u (u^{2} - 25)^{4} d u

Apply u-substitution

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

= \frac{1}{124416} \cdot \int \frac{v ^{4}}{2} d v

Take the constant out:

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

= \frac{1}{124416} \cdot \frac{1}{2} \cdot \int v^{4} d v

Apply the Power Rule

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

= \frac{1}{124416} \cdot \frac{1}{2} \cdot \frac{v ^{5}}{5}

Substitute back

= \frac{1}{124416} \cdot \frac{1}{2} \cdot \frac{( ( 6 x + 5 ) ^{2} - 25 ) ^{5}}{5}

Simplify

\frac{1}{124416} \cdot \frac{1}{2} \cdot \frac{( ( 6 x + 5 ) ^{2} - 25 ) ^{5}}{5} : \frac{1}{1244160} (36 x^{2} + 60 x)^{5}

= \frac{1}{1244160} (36 x^{2} + 60 x)^{5}

Add a constant to the solution

= \frac{1}{1244160} (36 x^{2} + 60 x)^{5} + C

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What is the integral of (3x^2+5x)^4(6x+5) ?
The integral of (3x^2+5x)^4(6x+5) is 1/1244160 (36x^2+60x)^5+C