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  4. The value of ∫(2+ sin x/1+ cos x) ex/2dx is

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Q. The value of $ \int{\frac{2+\sin x}{1+\cos x}}\,{{e}^{x/2}}dx $ is

J & K CETJ & K CET 2013Integrals

Solution:

Let $ l=\int{\frac{2+\sin x}{1+\cos x}}.\,\,{{e}^{x/2}}\,dx $
$ \Rightarrow $ $ l=\int{\frac{2+\frac{2\,\tan \,x/2}{1+{{\tan }^{2}}x/2}}{1+\frac{1-{{\tan }^{2}}x/2}{1+{{\tan }^{2}}x/2}}}\,.\,\,{{e}^{-x/2}}dx $
$ \Rightarrow $ $ l=\frac{2{{\tan }^{2}}\frac{x}{2}+2+2\tan \frac{x}{2}}{1+{{\tan }^{2}}\frac{x}{2}-{{\tan }^{2}}\frac{x}{2}+1}\,\,.\,{{e}^{x/2}}\,dx $
$ \Rightarrow $ $ l=2\int{\frac{{{\tan }^{2}}\frac{x}{2}+\tan \frac{x}{2}+1}{2}}.{{e}^{x/2}}\,\,dx $
$ \Rightarrow $ $ l=\int{{{\tan }^{2}}\frac{x}{2}.{{e}^{x/2}}dx+\int{\tan \frac{x}{2}.{{e}^{x/2}}\,dx}} $
$ +\int{{{e}^{x/2}}\,\,dx} $
$ \Rightarrow $ $ l=\int{\underset{II}{\mathop{{{\sec }^{2}}}}\,}\,x/2.{{e}^{x/2}}\,dx $
$ -\int{{{e}^{x/2}}\,dx+\int{\tan \frac{x}{2}.{{e}^{x/2}}\,dx+\int{{{e}^{x/2}}\,dx}}} $
$ \Rightarrow $ $ l=2{{e}^{x/2}}.\tan \frac{x}{2}-\int{\frac{1}{2}{{e}^{x/2}}.\tan \frac{x}{2}.2dx} $
$ +\int{\tan \,\frac{x}{2}.\,{{e}^{x/2}}\,dx+C} $
$ \Rightarrow $ $ l=2{{e}^{x/2}}.\tan \frac{x}{2}-\int{{{e}^{x/2}}.\tan \frac{x}{2}\,dx} $
$ +\int{{{e}^{x/2}}.\tan \frac{x}{2}dx+C} $
$ \Rightarrow $ $ l=2{{e}^{x/2}}.\tan \frac{x}{2}+C $