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Question
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Options
–1
0
1
2
Solution
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is 0.
Explanation:
f(x) = `e^(cos^2x) sin^3 (2n + 1)x`
f(–x) = `e^(cos^2(-x)) sin^3 (2n + 1)(-x)`
f(–x) = `-e^(cos^2x) sin^3 (2n + 1)x`
∵ f(–x) = –f(x)
So, `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` = 0
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