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

The integral from 0 to 4 of x^2e^{x/2} is 16e^2-16

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

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

16 e^{2} - 16

Decimal

102.22489 \dots

Solution steps

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

Apply Integration By Parts

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

\int 4 e^{\frac{x}{2}} x d x = 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})

= [2 e^{\frac{x}{2}} x^{2} - 16 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})]_{0}^{4}

Compute the boundaries:

16 e^{2} - 16

= 16 e^{2} - 16

\int_{5}^{8} 2400 x 3 + 2 x d x

\int_{0}^{\frac{1}{2}} e^{2 x} d x

\int_{0}^{2 π} - 4 cos^{2} (t) d t

\int_{3}^{5} \frac{x}{x ^{2} + 4 x + 13} d x

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

What is the integral from 0 to 4 of x^2e^{x/2} ?
The integral from 0 to 4 of x^2e^{x/2} is 16e^2-16