Question

Find $\int \frac{(3\sin \theta -2)\cos \theta }{5-{\cos }^2\theta -4\sin \theta }d\theta$.

• A. $I=-\ln |\sin \theta -2|-\frac{4}{\sin \theta -2}+C$
• B. $I=3\ln |\sin \theta -2|-\frac{4}{\sin \theta -2}+C$
• C. $I=3\ln |\cos \theta -2|-\frac{4}{\sin \theta -2}+C$
• D. None of these

Answer 1

Answer: (B)

$I=3\ln |\sin \theta -2|-\frac{4}{\sin \theta -2}+C$

$\int \frac{(3\sin \theta -2)\cos \theta }{5-{\cos }^2\theta -4\sin \theta }d\theta$

$\int \frac{(3\sin \theta -2)\cos \theta }{{\sin }^2\theta -4\sin \theta +4}d\theta$

Putting $\sin \theta =m$

$\cos \theta d\theta =dm$

$\int \frac{3m-2}{m^2-4m+4}dm$

$\int \frac{3(m-2)+4}{(m-2{)}^2}dm$

$\int \frac{3}{m-2}dm+\int \frac{4}{(m-2{)}^2}dm$

$3\ln |m-2|-\frac{4}{m-2}+C$

$I=3\ln |\sin \theta -2|-\frac{4}{\sin \theta -2}+C$