Question

Find \(\int_0^1 20x^3 e^{x^4}\) dx.

(a) (e-1)

(b) 5(e+1)

(c) 5e

(d) 5(e-1)

I had been asked this question during an interview.

My question comes from Evaluation of Definite Integrals by Substitution in portion Integrals of Mathematics – Class 12

Answer 1

Right option is (d) 5(e-1)

To explain I would say: I=\(\int_0^1 20x^3 e^{x^4}\) dx

Let x^4=t

Differentiating w.r.t x, we get

4x^3 dx=dt

∴The new limits

When x=0, t=0

When x=1,t=1

∴\(\int_0^1 \,20x^3 \,e^{x^4} \,dx=\int_0^1 5e^t dt\)

\(=5[e^t]_0^1=5(e^1-e^0)\)=5(e-1).