Evaluate the following Integral: ∫(1)/(3 + 2 sin x + cos x)dx, x ∈ [0, π]

Question

Evaluate the following Integral:

\(\int\limits_0^\pi\cfrac{1}{3+2sin\text x+cos\,\text x}d\text x \)

Answer 1

Given Definite Integral can be written as:

⇒ I(x) = \(\int\limits_0^\pi\cfrac{1}{3+2sin\text x+cos\,\text x}d\text x \)

We know that:

We know that: 1 + tan²x = sec²x

Let us assume,

y = 1 + tan\((\cfrac{\text x}2)\)

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

Upper limit for the Definite Integral:

Lower limit for the Definite Integral:

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

We know that:

We know that:

[here f’(x) is derivative of f(x)).