# How do you integrate #int xtan(x^2)sec(x^2)# using substitution?

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To integrate (\int x\tan(x^2)\sec(x^2)) using substitution, let (u = x^2). Then, (du = 2x dx) or (x dx = \frac{1}{2} du).

The integral becomes: [\int x\tan(x^2)\sec(x^2) dx = \frac{1}{2} \int \tan(u) \sec(u) du]

Now, recall that (\tan(u)\sec(u) = \sec(u)\tan(u)). So, the integral becomes: [\frac{1}{2} \int \sec(u) \tan(u) du = \frac{1}{2} \sec(u) + C]

Finally, substitute back (u = x^2) to get the final result: [\frac{1}{2} \sec(x^2) + C]

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