What is the integral from 0 to 2 of x^3e^{4x} ?

The integral from 0 to 2 of x^3e^{4x} is (362e^8+6)/(256)

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integral from 0 to 2 of x^3e^{4x}

\int_{0}^{2} x^{3} e^{4 x} d x

\frac{362 e ^{8} + 6}{256}

Decimal

4215.28434 \dots

Solution steps

\int_{0}^{2} x^{3} e^{4 x} d x

Apply u-substitution

= \int_{0}^{8} \frac{e ^{u} u ^{3}}{256} d u

Take the constant out:

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

= \frac{1}{256} \cdot \int_{0}^{8} e^{u} u^{3} d u

Apply Integration By Parts

= \frac{1}{256} [u^{3} e^{u} - \int 3 u^{2} e^{u} d u]_{0}^{8}

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= \frac{1}{256} [u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))]_{0}^{8}

Compute the boundaries:

362 e^{8} + 6

= \frac{1}{256} (362 e^{8} + 6)

Simplify

= \frac{362 e ^{8} + 6}{256}

What is the integral from 0 to 2 of x^3e^{4x} ?
The integral from 0 to 2 of x^3e^{4x} is (362e^8+6)/(256)