Question

please solve.

it's important
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Answer 1

Integrating the secant requires a bit of manipulation.

Multiply secx by secx+tanxsecx+tanx, which is really the same as multiplying by 1. Thus, we have

∫(secx(secx+tanx)secx+tanx)dx

∫sec2x+secxtanxsecx+tanxdx

Now, make the following substitution:

u=secx+tanx

du=(secxtanx+sec2x)dx=(sec2x+secxtanx)dx

We see that du appears in the numerator of the integral, so we may apply the substitution:

∫duu=ln|u|+C

Rewrite in terms of x to get

∫secxdx=ln|secx+tanx|+C

This is an integral worth memorizing.