Find \int_{-1}^1 \,2xe^x \,dx.

Category: QuestionsFind \int_{-1}^1 \,2xe^x \,dx.
Editor">Editor Staff asked 11 months ago

Find \int_{-1}^1 \,2xe^x \,dx.
 
(a) \frac{4}{e}
 
(b) 4e
 
(c) –\frac{4}{e}
 
(d) -4e
 
This question was posed to me in my homework.
 
I would like to ask this question from Fundamental Theorem of Calculus-1 in section 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

The correct choice is (a) \frac{4}{e}
 
To elaborate: I=\int_{-1}^1 \,2xe^x \,dx
 
F(x)=\int 2xe^x dx
 
By using the formula, \int u.v \,dx=u \int v \,dx-\int u'(\int v \,dx)
 
F(x)=2x\int e^x dx-\int(2x)’\int e^x \,dx
 
=2xe^x-\int 2e^x dx
 
=2e^x (x-1)
 
Therefore, by using the fundamental theorem of calculus, we get
 
I=F(1)-F(-1)
 
I=2e^1 (1-1)-2e^-1 (-1-1)
 
I=\frac{4}{e}.