What is the integral of (2x+1)(x^2+x)^4 ?

The integral of (2x+1)(x^2+x)^4 is 1/5120 (4x^2+4x)^5+C

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integral of (2x+1)(x^2+x)^4

\int (2 x + 1) (x^{2} + x)^{4} d x

\frac{1}{5120} (4 x^{2} + 4 x)^{5} + C

Solution steps

\int (2 x + 1) (x^{2} + x)^{4} d x

Apply u-substitution

: \int \frac{u ( u ^{2} - 1 ) ^{4}}{512} d u

= \int \frac{u ( u ^{2} - 1 ) ^{4}}{512} d u

Take the constant out:

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

= \frac{1}{512} \cdot \int u (u^{2} - 1)^{4} d u

Apply u-substitution

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

= \frac{1}{512} \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}{512} \cdot \frac{1}{2} \cdot \int v^{4} d v

Apply the Power Rule

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

= \frac{1}{512} \cdot \frac{1}{2} \cdot \frac{v ^{5}}{5}

Substitute back

= \frac{1}{512} \cdot \frac{1}{2} \cdot \frac{( ( 2 x + 1 ) ^{2} - 1 ) ^{5}}{5}

Simplify

\frac{1}{512} \cdot \frac{1}{2} \cdot \frac{( ( 2 x + 1 ) ^{2} - 1 ) ^{5}}{5} : \frac{1}{5120} (4 x^{2} + 4 x)^{5}

= \frac{1}{5120} (4 x^{2} + 4 x)^{5}

Add a constant to the solution

= \frac{1}{5120} (4 x^{2} + 4 x)^{5} + C

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What is the integral of (2x+1)(x^2+x)^4 ?
The integral of (2x+1)(x^2+x)^4 is 1/5120 (4x^2+4x)^5+C