What is the integral from 0 to 1 of 4(2x+1)^3 ?

The integral from 0 to 1 of 4(2x+1)^3 is 40

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integral from 0 to 1 of 4(2x+1)^3

\int_{0}^{1} 4 (2 x + 1)^{3} d x

40

Solution steps

\int_{0}^{1} 4 (2 x + 1)^{3} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{1} (2 x + 1)^{3} d x

Apply u-substitution

= 4 \cdot \int_{1}^{3} \frac{u ^{3}}{2} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int_{1}^{3} u^{3} d u

Apply the Power Rule

= 4 \cdot \frac{1}{2} [\frac{u ^{4}}{4}]_{1}^{3}

Simplify

4 \cdot \frac{1}{2} [\frac{u ^{4}}{4}]_{1}^{3} : 2 [\frac{u ^{4}}{4}]_{1}^{3}

= 2 [\frac{u ^{4}}{4}]_{1}^{3}

Compute the boundaries:

20

= 2 \cdot 20

Simplify

= 40

What is the integral from 0 to 1 of 4(2x+1)^3 ?
The integral from 0 to 1 of 4(2x+1)^3 is 40