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

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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}.