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Question
Hence evaluate:
`int_(-2π)^(2π) (sin^4x + cos^4x)/(1 + e^x)dx`
Solution
I = `int_(-2π)^(2π) (sin^4x + cos^4x)/(1 + e^x)dx`
= `int_(-2π)^(2π) (sin^4(-x) + cos^4(-x))/(1 + e^-x)dx` ...`[∵ int_a^b f(x)dx = int_a^b f(a + b - x)dx]`
= `int_(-2π)^(2π) (sin^4x + cos^4x)/(1 + e^-x)dx`
= `int_(-2π)^(2π) (e^x(sin^4x + cos^4x))/(1 + e^x)dx`
I + I = `int_(-2π)^(2π) (sin^4x + cos^4x)/(1 + e^x)dx + int_(-2π)^(2π) (e^x(sin^4x + cos^4x))/(1 + e^x)dx`
∴ 2I = `int_(-2π)^(2π) (sin^4x + cos^4x)dx`
= `2int_0^(2π) (sin^4x + cos^4x)dx` ...`[∵ int_-a^a f(x)dx = 2int_0^a f(x)dx "if" f(-x) = f(x)]`
∴ I = `int_0^(2π) (sin^4x + cos^4x)dx`
= `2int_0^π (sin^4x + cos^4x)dx` ...`[∵ int_0^(2a) f(x)dx = 2int_0^af(x)dx "if" f(2a - x) = f(x)]`
= `2*(3π)/4`
= `(3π)/2`
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