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

The integral of (2x+2)^2 is 1/6 (2x+2)^3+C

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integral of (2x+2)^2

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

\frac{1}{6} (2 x + 2)^{3} + C

Solution steps

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

Apply u-substitution

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

= \int \frac{u ^{2}}{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^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{1}{2} \cdot \frac{u ^{3}}{3}

Substitute back

u = 2 x + 2

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

Simplify

\frac{1}{2} \cdot \frac{( 2 x + 2 ) ^{3}}{3} : \frac{1}{6} (2 x + 2)^{3}

= \frac{1}{6} (2 x + 2)^{3}

Add a constant to the solution

= \frac{1}{6} (2 x + 2)^{3} + C

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