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 :
\[I=\int_{{}}^{{}}{\frac{\sin xdx}{3+4{{\cos }^{2}}x}}\cos x=t\] \[\Rightarrow -\sin x\,dx=dt\] \[I=-\int_{{}}^{{}}{\frac{dt}{3+4{{t}^{2}}}\Rightarrow I=}-\frac{1}{4}\int_{{}}^{{}}{\frac{dt}{\frac{3}{4}+{{t}^{2}}}}\] \[I=-\frac{1}{4}\int_{{}}^{{}}{\frac{dt}{{{\left( \frac{\sqrt{3}}{2} \right)}^{2}}+{{(t)}^{2}}}}\] \[=-\frac{1}{4}.\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{t}{\sqrt{3}/2} \right)+c\] \[=-\frac{1}{2\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\cos x}{\sqrt{3}} \right)+c\]
You need to login to perform this action.
You will be redirected in
3 sec
