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)
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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=\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