Question

Evaluate: $\int\limits^1_0 {{e^{x}}^{2} \cdotp x^{3}} \, dx$

Answer 1

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hmmm

Answer 2

we have to integrate the , $\int\limits^1_0{e^{x^2}.x^3}\,dx$

you can find it easily with the help of substitution.

let x² = p

differentiating both sides,

2x dx = dp

lower limit, p = (0)² = 0

upper limit , p = (1)² = 1

so, $\int\limits^1_0{e^{x^2}x.x^2}\,dx=\int\limits^1_0{e^p.p dp/2}$

= $\frac{1}{2}\int\limits^1_0{e^p.p}\,dp$

now applying integration by part,

$\int{f_1(x)f_2(x)}\,dx=f_1(x).\int{f_2(x)}\,dx-\int{f'_1(x).\int{f_2(x)}}\,dx$

so, I = ∫e^p.p dp = p ∫e^p dp - ∫(1) (∫e^p dp) dp

= p . e^p - e^p

= (p - 1)e^p

so, $\frac{1}{2}\int\limits^1_0{e^p.p}\,dp=\frac{1}{2}[(p-1)e^p]^1_0$

= 1/2