MCQ
$\int {} $ $e^{\tan \theta} $ $(\sec \theta - \sin \theta )$ $ d\theta $ equals :
- A$- e^{\tan \theta} \sin \theta + c$
- B$e^{\tan \theta}\sin \theta + c$
- C$e^{\tan \theta } \sec \theta + c$
- ✓$e^{\tan \theta} \cos \theta + c$
Substitute $\tan \theta=t \Rightarrow \sec ^{2} \theta d \theta=d t$
$\therefore I=\int e^{t}(\cos \theta-\tan \theta \sec \theta) d t$
$=\int e^{t}\left(\frac{1}{\sqrt{1+t^{2}}}-t \sqrt{1+t^{2}}\right) d t$
As $\frac{d}{d x}\left(\frac{1}{\sqrt{1+t^{2}}}\right)=t \sqrt{1+t^{2}}$
$\therefore I=e^{t}\left(\frac{1}{\sqrt{1+t^{2}}}\right)=e^{\tan \theta} \cos \theta+c$
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