∫ ( 1 + √ t a n x ) ( 1 + t a n 2 x ) 2 t a n x d x equal to Options: A . &nbsplogtan2x+√tanx+c B . &nbsplogtan2x+12√tanx+c C . &nbsplog|tanx|+2√tanx+c D . &nbsplog|tanx|+√tanx+c

Answer: Option A : A ∫ 1 2 s i n x c o s x d x + 1 2 ∫ √ t a n x s i n x c o s x d x = 1 2 ∫ s i n 2 x + c o s 2 x s i n x c o s x d x + 1 2 ∫ s e c 2 x √ t a n x d x = l o g ( t a n 2 x ) + √ t a n x + c

∫(1+√tanx)(1+tan2x)2tanxdx equal to

Question

\int \frac{(1 + \sqrt{t a n x}) (1 + t a n^{2} x)}{2 t a n x} d x

equal to

Options:

A . logtan2x+√tanx+c

B . logtan2x+12√tanx+c

C . log|tanx|+2√tanx+c

D . log|tanx|+√tanx+c

Answer: Option A
:
A

\int \frac{1}{2 s i n x c o s x} d x + \frac{1}{2} \int \frac{\sqrt{t a n x}}{s i n x c o s x} d x = \frac{1}{2} \int \frac{s i n^{2} x + c o s^{2} x}{s i n x c o s x} d x + \frac{1}{2} \int \frac{s e c^{2} x}{\sqrt{t a n x}} d x = l o g (t a n^{2} x) + \sqrt{t a n x} + c

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