Evaluate the following Integrals:
Let
Put x = atan2θ
⇒ x = 2a tan θ sec2θ dθ (Differentiating both sides)
When x = 0, atan2θ = 0 ⇒ tan θ = 0 ⇒ θ = 0
So, the new limits are 0 and.
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
So, the new limits are 0 and 1.
Substituting this in the original integral,
We will use integration by parts.
Recall
Here, take f(t) = tan-1t and g(t) = t
Now,
Substituting these values, we evaluate the integral.
We can write
Recall