What is the integral of (2x^2+2x-3)^{10}(2x+1) ?

The integral of (2x^2+2x-3)^{10}(2x+1) is 1/22 (2x^2+2x-3)^{11}+C

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integral of (2x^2+2x-3)^{10}(2x+1)

\int (2 x^{2} + 2 x - 3)^{10} (2 x + 1) d x

\frac{1}{22} (2 x^{2} + 2 x - 3)^{11} + C

Solution steps

\int (2 x^{2} + 2 x - 3)^{10} (2 x + 1) d x

Apply u-substitution

: \int \frac{u ^{10}}{2} d u

= \int \frac{u ^{10}}{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 u^{10} d u

Apply the Power Rule

: \frac{u ^{11}}{11}

= \frac{1}{2} \cdot \frac{u ^{11}}{11}

Substitute back

u = 2 x^{2} + 2 x - 3

= \frac{1}{2} \cdot \frac{( 2 x ^{2} + 2 x - 3 ) ^{11}}{11}

Simplify

\frac{1}{2} \cdot \frac{( 2 x ^{2} + 2 x - 3 ) ^{11}}{11} : \frac{1}{22} (2 x^{2} + 2 x - 3)^{11}

= \frac{1}{22} (2 x^{2} + 2 x - 3)^{11}

Add a constant to the solution

= \frac{1}{22} (2 x^{2} + 2 x - 3)^{11} + C

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What is the integral of (2x^2+2x-3)^{10}(2x+1) ?
The integral of (2x^2+2x-3)^{10}(2x+1) is 1/22 (2x^2+2x-3)^{11}+C