Question

Evaluate
8
x ²(1 – e-*) dx
0​

Answer 1

Correct Question:-

Evaluate $\displaystyle\int\limits_0^8x^2\left(1-e^{-x}\right)\ dx.$

Solution:-

We're asked to evaluate the integral,

$\displaystyle\longrightarrow I=\int\limits_0^8x^2\left(1-e^{-x}\right)\ dx$

$\displaystyle\longrightarrow I=\int\limits_0^8\left(x^2-x^2e^{-x}\right)\ dx$

$\displaystyle\longrightarrow I=\int\limits_0^8x^2\ dx-\int\limits_0^8x^2e^{-x}\right)\ dx\quad\quad\dots(1)$

Consider,

$\displaystyle\longrightarrow I_1=\int\limits_0^8x^2\ dx$

$\displaystyle\longrightarrow I_1=\dfrac{1}{3}\left[x^3\right]_0^8$

$\displaystyle\longrightarrow I_1=\dfrac{512}{3}$

Consider,

$\displaystyle\longrightarrow I_2=\int\limits_0^8x^2e^{-x}\ dx$

$\displaystyle\longrightarrow I_2=-\left[(x^2-2x+2)e^{-x}\right]_0^8$

$\displaystyle\longrightarrow I_2=-\left(50e^{-8}-2\right)$

$\displaystyle\longrightarrow I_2=2-50e^{-8}$

Then (1) becomes,

$\displaystyle\longrightarrow I=\dfrac{512}{3}-\left(2-\dfrac{50}{e^8}\right)$

$\displaystyle\longrightarrow\underline{\underline{I=\dfrac{50}{e^8}+\dfrac{506}{3}}}$