Question

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

Answer 1

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=\(\frac{4}{e}\).