Step by step solution to "Evaluate for any integer, n∫−1212sin4(nπx)sin2(πx)"

navantegipowh

Answered question

2022-04-05

Evaluate for any integer, n

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sin^{4} (n π x)}{\sin^{2} (π x)}

Ishaan Stout

Beginner2022-04-06Added 14 answers

Let

α = 2 x

and

ϕ = x

then we have

\cos (x) + \cos (3 x) + \dots + \cos ((2 m + 1) x) = \cos (ϕ + 0 α) + \cos (ϕ + 1 α) + . . + \cos (ϕ + m α)

= \frac{\sin \frac{(m + 1) α}{2} \cos (ϕ + \frac{m α}{2})}{\sin \frac{α}{2}}

= \frac{\sin [(m + 1) x] \cos [(m + 1) x]}{\sin x}

= \frac{\sin [2 (m + 1) x]}{2 \sin x}

Therefore

\frac{\sin^{4} [2 (m + 1) x]}{\sin^{2} x} = 2^{4} \sin^{2} x {(\cos (x) + \cos (3 x) + \dots + \cos ((2 m + 1) x))}^{4}

Therefore

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{\sin^{4} [2 (m + 1) x]}{\sin^{2} x} d x = 2^{4} \int_{- \frac{1}{2}}^{\frac{1}{2}} \sin^{2} x {(\cos (x) + \cos (3 x) + \dots + \cos ((2 m + 1) x))}^{4} d x

The latter is tedious but doable for any value of m.

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