How do you find the antiderivative of #int sec^2xcsc^2x dx#?

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Answer 1 · 2017-02-16T05:07:11

#tanx - cotx + C#

Do a little bit of experimentation using some trig identities. Recall the pythagorean identity #color(red)(csc^2alpha = 1 + cot^2alpha)#.

#=intsec^2x(1 + cot^2x)dx#

Now recall that cotangent function is the reciprocal of the tangent function and the secant function is the reciprocal of the cosine function.

#=int1/cos^2x(1 + cos^2x/sin^2x)dx#

#=int 1/cos^2x + 1/sin^2xdx#

#=int sec^2x + csc^2xdx#

#=intsec^2x + intcsc^2xdx#

These are both widely known integrals. If you haven't already, I would recommend learning them by heart.

#=tanx - cotx + C#

Hopefully this helps!