For x2 nπ + 1, nN (the set of natural numbers), the integral is equal to (where c is a constant of integration)

Engineering

Mathematics

Integration of Trigonometric Functions

Question

For x² $\neq$ nπ + 1, n $\in$ N (the set of natural numbers), the integral $\int x \sqrt{\frac{2 \sin (x^{2} - 1) - \sin 2 (x^{2} - 1)}{2 \sin (x^{2} - 1) + \sin 2 (x^{2} - 1)}} dx$ is equal to (where c is a constant of integration)

$\log_{e} | \sec (\frac{x^{2} - 1}{2}) | + c$

$\frac{1}{2} \log_{e} | \sec^{2} (\frac{x^{2} - 1}{2}) | + c$

$\frac{1}{2} \log_{e} | \sec (x^{2} - 1) | + c$

$\log_{e} | \frac{1}{2} \sec^{2} (x^{2} - 1) | + c$

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$\int x \sqrt{\frac{2 \sin (x^{2} - 1) (1 - \cos (x^{2} - 1))}{2 \sin (x^{2} - 1) (1 + \cos (x^{2} - 1))}}$

$= \int x \frac{\sin (\frac{x^{2} - 1}{2})}{\cos (\frac{x^{2} - 1}{2})} dx$

$= \int x \tan (\frac{x^{2} - 1}{2}) dx$

$Let \frac{x^{2} - 1}{2} = t \Rightarrow 2 xdx = 2 dt$

$= \int \tan (t) dt = ℓn | \sec t | + c$

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