Definite Integration Question 38
Question 38
- For any integer $n$, the integral $\int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}(2 n+1) x d x$ has the value
(1985, 2M) (a) $\pi$ (b) 1 (c) 0 (d) None of these
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Solution:
- Let $I=\int_{0}^{\pi} e^{\cos ^{2} x} \cdot \cos ^{3}{(2 n+1) x} d x$
$$ 0, \quad f(a-x)=-f(x) $$
Again, let $f(x)=e^{\cos ^{2} x} \cdot \cos ^{3}{(2 n+1) x}$
$\therefore \quad f(\pi-x)=\left(e^{\cos ^{2} x}\right)\left{-\cos ^{3}(2 n+1) x\right}=-f(x)$
$$ \therefore \quad I=0 $$
