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 
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1 Answers
Editor">Editor Staff answered 11 months ago

Right option is (d) 5(e-1)
 
To explain I would say: I=

1
0
20
x
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
 


1
0
20
x
3
e
x
4
dx=

1
0
5
e
t
dt
 
 
=5[
e
t
]
1
0
=5(
e
1

e
0
)
=5(e-1).