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

Question

Evaluate the following Integral:

\(\int\limits_{0}^{\pi/2}\cfrac{sin \text x\,cos\,\text x}{cos^2\text x+3\,cos\,\text x+2}d \text x\)

Answer 1

Let I = \(\int\limits_{0}^{\pi/2}\cfrac{sin \text x\,cos\,\text x}{cos^2\text x+3\,cos\,\text x+2}d \text x\)

In the denominator, we can write

cos²x + 3 cos x + 2 = (cos x + 1)(cos x + 2)

Put cos x = t

⇒ –sin(x)dx = dt (Differentiating both sides)

⇒ sin(x)dx = –dt

When x = 0, t = cos 0 = 1

When x = \(\cfrac{\pi}2\), t = cos \(\cfrac{\pi}2\) = 0.

So, the new limits are 1 and 0.

Substituting this in the original integral,

Recall

⇒ I = – [2(ln|0+2| – ln|1+2|) – (ln|0+1| – ln|1+1|)]

⇒ I = – [2(ln 2 – ln 3) – (ln 1 – ln 2)]

⇒ I = – (2 ln 2 – 2 ln 3 – 0 + ln 2)

⇒ I = – (3 ln 2 – 2 ln 3)

⇒ I = 2 ln 3 – 3 ln 2

⇒ I = ln 9 – ln 8 = ln\(\cfrac98\)