Keep calm and Integrate

Calculus Level 4

$\int_{\pi/4}^{\pi/3}12\csc (2x)[\ln(\sin^2(x))-\ln(\cos^2(x))]^2 \, dx$

If the above integral equals $(\ln b)^a$ , where $a$ and $b$ are positive integers, find $a+b$ .

1 solution

Harsh Shrivastava
Feb 23, 2016

$K = \displaystyle \int 12\csc (2x)[ln(sin^2(x))-ln(cos^2(x))]^2 dx$

$ln(sin^2(x))-ln(cos^2(x)) = \ln \tan ^{2} x$

Thus our integrand becomes

$K =\displaystyle \int 12\csc (2x)[\ln \tan ^{2} x]^2 dx$

Now let $t = ln \tan ^{2} x$

$\frac{dt}{dx} = 4\csc 2x$

Therefor $K = \displaystyle\int 3t^{2} dt$

Which on evaluating and substituting back $t = \ln \tan^{2} x$ gives $K = \ln ^{3} \tan ^{2} x$

On putting limits and evaluating, final answer becomes $\ln ^{3} 3$