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

The integral of (x^3+1)^{10}x^2 is 1/33 (x^3+1)^{11}+C

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

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

\frac{1}{33} (x^{3} + 1)^{11} + C

Solution steps

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

Apply u-substitution

= \int \frac{u ^{10}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{10} d u

Apply the Power Rule

= \frac{1}{3} \cdot \frac{u ^{11}}{11}

Substitute back

u = x^{3} + 1

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

Simplify

\frac{1}{3} \cdot \frac{( x ^{3} + 1 ) ^{11}}{11} : \frac{1}{33} (x^{3} + 1)^{11}

= \frac{1}{33} (x^{3} + 1)^{11}

Add a constant to the solution

= \frac{1}{33} (x^{3} + 1)^{11} + C

What is the integral of (x^3+1)^{10}x^2 ?
The integral of (x^3+1)^{10}x^2 is 1/33 (x^3+1)^{11}+C