Evaluate the following Integral: ∫(cos x)/(3 + 4 sin x)dx, x ∈ [0, π/3]

Question

Evaluate the following Integral:

\(\int\limits_0^{\pi/3}\cfrac{cos\text x}{3+4sin\text x}d\text x \)

Answer 1

Given Definite Integral can be written as:

⇒ I(x) = \(\int\limits_0^{\pi/3}\cfrac{cos\text x}{3+4sin\text x}d\text x \)....(1)

Let us assume 3 + 4sinx = y

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

⇒ d(3 + 4sinx) = d(y)

⇒ 4cosx dx = dy

⇒ cos x dx = \(\cfrac{dy}4\).......(2)

Lower limit for the Definite Integral:

⇒ x = 0 ⇒ y = 3 + 4sin(0)

⇒ y = 3 + 0

⇒ y = 3……(3)

Upper limit for the Definite Integral:

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

We know that:

\(\int\cfrac{d\text x}{\text x}\) = log x + c

 We know that:

[here f’(x) is derivative of f(x))