Evaluate the following Integrals:
Let
Put cos-1x = t
⇒ x = cos t
⇒ dx = – sin t dt (Differentiating both sides)
When x = 1, t = cos-1(1) = 0
So, the new limits are and 0.
Substituting this in the original integral,
We will use integration by parts.
Recall
Here, take f(t) = t2 and g(t) = sin t
Now,
⇒ f’(t) = 2t
Substituting these values, we evaluate the integral.
Let
We use integration by parts again.
Here, take f(t) = t and g(t) = cos t
Now,
⇒ f’(t) = 1
Using these values in equation for I1
Substituting I1 in I, we get
⇒ I = –2 – (–π) = π – 2