Definite Integrals — Maths STD 12 Science — Question

4. $\log\Big(\frac{4}{3}\Big)$

Solution:

Let, $\text{I}=\int\limits^\frac{\pi}{2}_0\frac{\cos\text{x}}{(2+\sin\text{x})(1+\sin\text{x})}\text{dx}$

Let $\sin\text{x}$ then $\cos\text{x}\text{ dx}=\text{dt}$

When $\text{x}=0,\text{t}=0,\text{x}=\frac{\pi}{2},\text{t}=1$

Therefore the integral becomes

$\text{I}=\int\limits^1_0\frac{\text{dt}}{(2+\text{t})(1+\text{t})}$

$=\int\limits^1_0\Big[\frac{-1}{2+\text{t}}+\frac{1}{1+\text{t}}\Big]\text{dt}$

$=\big[-\log(2+\text{t})+\log(1+\text{t})\big]^1_0$

$= \big[\log(1+\text{t})-\log(2+\text{t})\big]^1_0$

$=\log2-\log3-\log1+\log2$

$=\log\frac{4}{3}$