Evaluate the following Integral: ∫ (xsin^-1)/(√(1 - x^2))dx, x ∈ [0, 1/2]

Question

Evaluate the following Integral:

\(\int\limits_0^{1/2}\cfrac{\text xsin^{-1}}{\sqrt{1-\text x^2}}d\text x \)

Answer 1

Given Definite Integral can be written as:

⇒ I(x) = \(\int\limits_0^{1/2}\cfrac{\text xsin^{-1}}{\sqrt{1-\text x^2}}d\text x \)

Let us find the value of \(\int\limits_0^{1/2}\cfrac{\text xsin^{-1}}{\sqrt{1-\text x^2}}d\text x \) using by parts integration,

Now we substitute this result in the Definite Integral:

We know that:

[here f’(x) is derivative of f(x)).