SATHEE: Indefinite Integration 1 Question 38

Indefinite Integration 1 Question 38

39. For any integer $n$ , the integral $\int_{0}^{π} e^{\cos^{2} x} \cos^{3} (2 n + 1) x d x$ has the value

(1985, 2M)

(a) $π$

(b) 1

(c) 0

(d) None of these

Show Answer

Solution:

Let $I = \int_{0}^{π} e^{\cos^{2} x} \cdot \cos^{3} (2 n + 1) x d x$

$0, f (a - x) = - f (x)$

Again, let $f (x) = e^{\cos^{2} x} \cdot \cos^{3} (2 n + 1) x$

$∴ f (π - x) = (e^{\cos^{2} x}) - \cos^{3} (2 n + 1) x = - f (x)$

$∴ I = 0$

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