Evaluate the integral(0 to 1/2). (use c for the constant of integration.) cos^-1(x) dx

Solution:

Given ∫cos^-1(x) dx

Rewrite it as ∫ 1.cos^-1(x) dx

Here, u = 1 and v = cos^-1(x)

using the integration of uv formula, we find the integral as follows:

We know that ∫ uv = u ∫ v + ∫ u’ ∫ v

∫\(_0 ^{½}\) 1.cos^-1(x) dx = xcos^-1(x) dx + ∫ x / √(1 - x²) dx + C

= [(xcos^-1(x) - √(1 - x²))]\(_0^{½}\)

= 1/2 cos^-1(1/2) - √(1 - 1/4) - (0 - 1)

= π/6 - √3/2 + 1

∫\(_0 ^{½}\) cos^-1(x) dx = π/6 - √3/2 + 1

Evaluate the integral(0 to 1/2). (use c for the constant of integration.) cos^-1(x) dx

Summary:

The integral(0 to 1/2). (use c for the constant of integration.) co^-1(x) dx is equal to π/6 - √3/2 + 1.