A) \[\log \tan \theta +{{\tan }^{2}}\theta +c\]
B) \[\log \tan \theta -\frac{1}{2}{{\tan }^{2}}\theta +c\]
C) \[\log \tan \theta +\frac{1}{2}{{\tan }^{2}}\theta +c\]
D) None of these
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\frac{d\theta }{\sin \theta {{\cos }^{3}}\theta }=\int_{{}}^{{}}{\frac{{{\sec }^{2}}\theta \,d\theta }{\sin \theta \cos \theta }=\int_{{}}^{{}}{\frac{{{\sec }^{2}}\theta (1+{{\tan }^{2}}\theta )}{\tan \theta }}}}\text{ }d\theta \] Put \[t=\tan \theta \Rightarrow dt={{\sec }^{2}}\theta \,d\theta ,\] then it reduces to \[\int_{{}}^{{}}{\frac{1+{{t}^{2}}}{t}\,dt=\int_{{}}^{{}}{\left( \frac{1}{t}+t \right)\,dt}}\] \[=\log t+\frac{{{t}^{2}}}{2}+c=\log \tan \theta +\frac{{{\tan }^{2}}\theta }{2}+c.\]
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