Question

$\int\limits^2_0 (eˣ-x)dx,Obtain the given integrals as the limit of a sum.$

Answer 1

HELLO DEAR,

$\sf{\int\limits^2_0 (e^x - x)\,dx}$

$\sf{\int\limits^2_0 (e^x.dx - x.dx)}$

$\sf{[e^x - x^2/2]^2_0}$

$\sf{[e^2 - e^0 - (4)/2]}$

$\sf{(e^2 - 1 - 2)}$

$\sf{(e^2 - 3)}$


I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

we have to get the value of $\int\limits^2_0{(e^x-x)}\,dx$

we know, $\int{e^x}\,dx=e^x+C$

$\int{x^n}\,dx=\left[\frac{x^{n+1}}{n+1}\right]+C$

$\int\limits^2_0{(e^x-x)}\,dx\\\\\\=\int\limits^2_0{e^x}\,dx-\int\limits^2_0{x}\,dx\\\\\\=[e^x]^2_0-\left[\frac{x^2}{2}\right]^2_0\\\\\\=(e^2-e^0)-\frac{1}{2}(2^2-0)\\\\\\=e^2-1-2=e^2-3$