Question

Evaluate the definite integral as limit of sums: $∫^1_0 xe^{x^2} dx$

Answer 1

$Let\, I=∫^1_0 xe^{x^2} dx$

$Put\, x^2=t⇒2x\, dx=dt$

$As\, x→0,t→0 \,and\, as\, x→1,t→1,$

$∴I=\frac{1}{2}∫^1_0e^t dt$

$\frac{1}{2}∫e^tdt=\frac{1}{2}e^t=F(t)$

By second fundamental theorem of calculus,we obtain

$I=F(1)-F(0)$

$=\frac{1}{2}e-\frac{1}{2}e°$

$=\frac{1}{2}(e-1)$