Evaluate the following Integral: ∫ (1)/(a^2sin^2 x + b^2 cos^2x)dx, x ∈ [0, π/2]

Question

Evaluate the following Integral:

\(\int\limits_0^{\pi/2}\cfrac{1}{a^2sin^2\text x+b^2cos^2\text x}d\text x \)

Answer 1

Given Definite Integral can be written as:

Let us assume y = tan x

Differentiating w.r.t x on both sides we get,

⇒ d(y) = d(tan x) ⇒ d y = sec²xdx……(2)

Upper limit for the Definite Integral:

Lower limit for the Definite Integral:

⇒ x = 0 ⇒ y = tan(0)

⇒ y = 0……(4)

Substituting (2),(3),(4) in the eq(1) we get,

[here f’(x) is derivative of f(x)).