A) \[\log (3+4{{\cos }^{2}}x)+c\]
B) \[\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{\cos x}{\sqrt{3}} \right)+c\]
C) \[\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\]
D) \[\frac{1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\]
Correct Answer: C
Solution :
\[I=\int{\frac{\sin x}{3+4{{\cos }^{2}}x}dx}\] Put \[\cos x=t\]Þ \[-\sin x\,\,dx=dt\] \[\therefore \,\,I=\int{\frac{-dt}{3+4{{t}^{2}}}}\]\[=\int{\frac{{{\sec }^{2}}x}{1-{{\tan }^{2}}x}dx}\] Þ \[I=-\frac{1}{4.\frac{\sqrt{3}}{2}}.{{\tan }^{-1}}\frac{t}{\left( \frac{\sqrt{3}}{2} \right)}+c=\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\frac{2\,t}{\sqrt{3}}+c\] \[\Rightarrow I=\frac{-1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\].
You need to login to perform this action.
You will be redirected in
3 sec
