Evaluate the following integral: ∫ x log sin x dx, x ∈ [0, π]

Question

Evaluate the following integral:

\(\int\limits_{0}^{\pi} \)x log sin x dx

Answer 1

Let us assume

I =  \(\int\limits_{0}^{\pi} \)x log sin x dx...equation 1

By property, we know that

Adding equation 1 and equation 2

= \(\int\limits_{0}^{a} \)f(x)dx if f(2a - x) = f(x)

= 0 if f(2a - x)= -f(x)

Thus equation 3 becomes

since log sin(π – x) = log sinx

By property, we know that

Adding equation 4 and equation 5

We know log m + log n = log m n

thus

since log(m/n) = log m – log n

Let I₁ = π\(\int\limits_{0}^{\pi/2} \)log sin 2x dx

Let 2x = y

2dx = dy

dx = dy/2

For x = 0

y = 0

for x = \(\cfrac{\pi}2\)

y = π

thus substituting value in I₁

Thus substituting the value of I₁ in equation 6