e^ ( - ) d equals : - Vidyadip

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
$e^{\tan \theta } \sec \theta + c$
✓
$e^{\tan \theta} \cos \theta + c$

✓

Answer

Correct option: D.

$e^{\tan \theta} \cos \theta + c$

d
Let $I=\int e^{\tan \theta}(\sec \theta-\sec \theta) d \theta$

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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