Question

For an integer n, the integral $\int_{0}^{\pi}e^{\cos^{2}x}$ cos³ (2n +1) x dx has the value

• A. π
• B. 1
• C. 0
• D. None of these

Answer 1

Answer: (C)

0

Answer 2

I = $\int_{0}^{\pi}e^{\cos^{2}x}$cos³ (2n +1) x dx

=$\int_{0}^{\pi}e^{\cos^{2}(\pi - x)}$cos³ (2 n + 1) (π – x) dx

= $\int_{0}^{\pi}e^{\cos^{2}x}$cos³ (2 n π + π – (2 n + 1) x) dx

= –$\int_{0}^{\pi}e^{\cos^{2}x}$cos³ (2n + 1) x dx = –I.

Hence 2 I = 0 ⇒ I = 0.