7.1 inderfinite integral — Maths STD 12 Science — Question

$\mathrm{I}_{1}=\int \frac{\cos \mathrm{x}}{\cos (\mathrm{x}-\mathrm{a})} \mathrm{dx} \Rightarrow \int \frac{\cos (\mathrm{x}-\mathrm{a}+\mathrm{a})}{\cos (\mathrm{x}-\mathrm{a})} \mathrm{dx}$

$\Rightarrow \int \frac{\cos (x-a) \cos a-\sin (x-a) \sin a}{\cos (x-a)} d x$

$\Rightarrow \cos a \int d x-\sin a \int \tan (x-a) d x$

$\mathrm{I}_{1} \Rightarrow \mathrm{x} \cos \mathrm{a}+\sin \mathrm{a} \log |\cos (\mathrm{x}-\mathrm{a})|+\mathrm{C}_{1} $         .......$(2)$

and $ \mathrm{I}_{2}=\int \frac{\sin \mathrm{x}}{\sin (\mathrm{x}-\mathrm{a})} \mathrm{d} \mathrm{x}$

$=\int \frac{\sin (x-a+a)}{\sin (x-a)} d x \Rightarrow \int \frac{\sin (x-a) \cos a+\cos (x-a) \sin a}{\sin (x-a)}$

$\Rightarrow \cos a \int d x+\sin a \int \cot (x-a) d x$

$I_{2}=x \cos a+\sin a \log |\sin (x-a)|+C_{2}$        ......$(3)$

by eq $(1)$

$\mathrm{I}=\mathrm{x} \cos \mathrm{a}+\sin \mathrm{a} \log \cos (\mathrm{x}-\mathrm{a})-\mathrm{x} \cos \mathrm{a}-$

$\sin a \log \sin (x-a)+C$

$=\sin a \log \left|\frac{\cos (x-a)}{\sin (x-a)}\right|+C$

$=\sin a \log |\cot (x-a)|+C$