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

Category: QuestionsFind \int_0^1 20x^3 e^{x^4} dx.
Editor">Editor Staff asked 11 months ago

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
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

1 Answers
Editor">Editor Staff answered 11 months ago

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).