Mathematics - Indefinite Integration Question with Solution | TestHub

Let $I=\int e^{\tan \theta }(\sec \theta -\sin \theta )d\theta$ Put $\tan \theta =t\Rightarrow {\sec }^2\theta d\theta =dt\Rightarrow d\theta =\frac{dt}{1+t^2}$ $\Rightarrow I=\int e^t\left(\sqrt{1+t^2}-\frac{t}{\sqrt{1+t^2}}\right)\frac{dt}{1+t^2}$ $=\int e^t\left(\frac{1}{\sqrt{1+t^2}}-\frac{t}{{\left(1+t^2\right)}^{3/2}}\right)dt$ Integrating first part by parts we have, $=\frac{1}{\sqrt{1+t^2}}{e}^t+\int \frac{t}{{\left(1+t^2\right)}^{3/2}}\cdot e^{t}dt$ $-\int \frac{t}{{\left(1+t^2\right)}^{3/2}}{e}^{t}dt+c$ $=\frac{e^t}{\sqrt{1+t^2}}+c=e^{\tan \theta }\cos \theta +c$