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

Question

Evaluate the following Integral:

\(\int\limits_{0}^{a}sin^{-1}\sqrt{\cfrac{\text x}{a+\text x}} d\text x\) 

Answer 1

Let I = \(\int\limits_{0}^{a}sin^{-1}\sqrt{\cfrac{\text x}{a+\text x}} d\text x\)

Put x = atan²θ

⇒ x = 2a tan θ sec²θ dθ (Differentiating both sides)

When x = 0, a tan²θ = 0

⇒ tan θ = 0 ⇒ θ = 0

When x = a, a tan²θ = a

⇒ tan θ = 1 ⇒ θ = \(\cfrac{\pi}4\)

So, the new limits are 0 and \(\cfrac{\pi}4\).

Also,

We have the trigonometric identity 1 + tan²θ = sec²θ

Substituting this in the original integral,

Now, put tan θ = t

⇒ sec²θ dθ = dt (Differentiating both sides)

When θ = 0, t = tan 0 = 0

When θ = \(\cfrac{\pi}4\), t = tan \(\cfrac{\pi}4\) = 1

So, the new limits are 0 and 1.

Substituting this in the original integral,

Substituting these values, we evaluate the integral.