Question

Question:-

$\rm \: {x}^{2} \not= n\pi + 1 \:, n \in \: N(the \: set \: of \: natural \: numers), \: the \:$
$\rm \int \: x \sqrt{ \dfrac{2 \sin( {x}^{2} - 1) - \sin2( {x}^{2} - 1 ) }{2 \sin( {x}^{2} - 1 ) + \sin( {x}^{2} - 1 ) } } dx$
( Where c is a constant of integration)
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Answer 1

Answer:

your answer attacted in the photo

Answer 2

put $\large\rm { ( x^{2} -1) = 1}$

$\large\rm { \implies 2x \ dx = dt}$

$\large\rm { \therefore I = \frac{1}{2} \displaystyle\int \rm { \sqrt{ \frac{1 - \cos \ t}{1 + \cos t}}} dt }$

$\large\rm { = \frac{1}{2} \int \tan \big ( \frac{t}{2} \big ) dt}$

$\large\rm { = \ln \bigg | \sec \frac{t}{2} \bigg | + c}$

$\large\rm { = \ln \bigg | \sec \big ( \frac{ x^{2} -1}{2} \big ) \bigg | + c}$

or

$\large\rm { \log_{e} ln \bigg | \sec \big ( \frac{ x^{2} -1}{2} \big ) \bigg | + c}$