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 . &nbsp l o g | 1 2 s e c ( x 2 + 1 ) | + C B . &nbsp l o g | s e c ( x 2 + 1 2 ) | + C C . &nbsp 1 2 l o g | s e c ( x 2 + 1 ) | + C D . &nbsp None of these

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

Discussion Forum : Indefinite Integration

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 .

l o g | \frac{1}{2} s e c (x^{2} + 1) | + C

B .

l o g | s e c (\frac{x^{2} + 1}{2}) | + C

C .

\frac{1}{2} l o g | s e c (x^{2} + 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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Discussion Forum : Indefinite Integration

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