What is the integral from 0 to 2 of 4x^2e^{2x} ?

The integral from 0 to 2 of 4x^2e^{2x} is 5e^4-1

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integral from 0 to 2 of 4x^2e^{2x}

\int_{0}^{2} 4 x^{2} e^{2 x} d x

5 e^{4} - 1

Decimal

271.99075 \dots

Solution steps

\int_{0}^{2} 4 x^{2} e^{2 x} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{2} x^{2} e^{2 x} d x

Apply Integration By Parts

: [\frac{1}{2} e^{2 x} x^{2} - \int e^{2 x} x d x]_{0}^{2}

= 4 [\frac{1}{2} e^{2 x} x^{2} - \int e^{2 x} x d x]_{0}^{2}

\int e^{2 x} x d x = \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})

= 4 [\frac{1}{2} e^{2 x} x^{2} - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x})]_{0}^{2}

Compute the boundaries:

2 e^{4} - \frac{3 e ^{4}}{4} - \frac{1}{4}

= 4 (2 e^{4} - \frac{3 e ^{4}}{4} - \frac{1}{4})

Simplify

= 5 e^{4} - 1

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What is the integral from 0 to 2 of 4x^2e^{2x} ?
The integral from 0 to 2 of 4x^2e^{2x} is 5e^4-1