Let x 2 ≠ n π − 1 , n ϵ N . Then, the value of ∫ x √ 2 s i n ( x 2 + 1 ) − s i n 2 ( x 2 + 1 ) 2 s i n ( x 2 + 1 ) + s i n 2 ( x 2 + 1 ) d x is equal to Options: A . &nbsplog|12sec(x2+1)|+C B . &nbsplog|sec(x2+12)|+C C . &nbsp12log|sec(x2+1)|+C D . &nbspNone of these

Let x2≠nπ−1,nϵN. Then, the value of ∫x

Question

Let

x^{2} \neq n π - 1, n ϵ N

. Then, the value of

\int x \sqrt{\frac{2 s i n (x^{2} + 1) - s i n 2 (x^{2} + 1)}{2 s i n (x^{2} + 1) + s i n 2 (x^{2} + 1)}} d x

is equal to

Options:

A . log|12sec(x2+1)|+C

B . log|sec(x2+12)|+C

C . 12log|sec(x2+1)|+C

D . None of these

Answer: Option B
:
B
We have,

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

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