What is the integral of (2x)^32 ?

The integral of (2x)^32 is 4x^4+C

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integral of (2x)^32

\int (2 x)^{3} 2 d x

4 x^{4} + C

Solution steps

\int (2 x)^{3} \cdot 2 d x

Take the constant out:

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

= 2 \cdot \int (2 x)^{3} d x

Simplify

(2 x)^{3} : 8 x^{3}

= 2 \cdot \int 8 x^{3} d x

Take the constant out:

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

= 2 \cdot 8 \cdot \int x^{3} d x

Apply the Power Rule

: \frac{x ^{4}}{4}

= 2 \cdot 8 \cdot \frac{x ^{4}}{4}

Simplify

2 \cdot 8 \cdot \frac{x ^{4}}{4} : 4 x^{4}

= 4 x^{4}

Add a constant to the solution

= 4 x^{4} + C

x \to 2 lim (\frac{1}{4 - x ^{2}})

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