Question

Find \(\int_{-1}^1 \,7x^6 \,(x^7+8)dx\)

(a) -386

(b) –\(\frac{386}{3}\)

(c) \(\frac{386}{3}\)

(d) 386

I have been asked this question in examination.

The question is from Evaluation of Definite Integrals by Substitution in section Integrals of Mathematics – Class 12

Answer 1

Correct answer is (c) \(\frac{386}{3}\)

For explanation I would say: I=\(\int_{-1}^1 \,7x^6 \,(x^7+8)dx\)

Let x^7+8=t

Differentiating w.r.t x, we get

7x^6 dx=dt

The new limits

When x=-1,t=7

When x=1,t=9

∴\(\int_{-1}^1 \,7x^6 \,(x^7+8)dx=\int_7^9 \,t^2 \,dt\)

=\([\frac{t^3}{3}]_7^9=\frac{1}{3} (9^3-7^3)=\frac{386}{3}\).