What is the integral of (x^2+2)^52x ?

The integral of (x^2+2)^52x is 1/6 (x^2+2)^6+C

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integral of (x^2+2)^52x

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

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

Solution steps

\int (x^{2} + 2)^{5} \cdot 2 x d x

Take the constant out:

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

= 2 \cdot \int (x^{2} + 2)^{5} x d x

Apply u-substitution

: \int \frac{u ^{5}}{2} d u

= 2 \cdot \int \frac{u ^{5}}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{6}}{6}

Substitute back

u = x^{2} + 2

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (x^2+2)^52x ?
The integral of (x^2+2)^52x is 1/6 (x^2+2)^6+C