How do I evaluiate intsec(x)(sec(x) + tan(x)) dx?

1 Answer
May 1, 2018

int secx(secx+tanx) dx = secx + tanx + c

Explanation:

I don't believe that there is any trigonometric substitution involved here actually.

The integrand secx(secx+tanx) can be expanded out to get sec^2x +secxtanx which are actually two simple functions to be antidifferentiated.

int sec^2x dx = tanx + c because d/dx(tanx) = sec^2x.

int secxtanx dx = secx which we will prove using substitution.

Expressing that in terms of sine and cosine, we get secxtanx = sinx/cos^2x so the integral becomes:

int sinx/cos^2x dx, to which we apply the substitution u=cosx.

u=cosx
:. (du)/dx=-sinx
:. -du = sinxdx

Now, using the change of variable rule, we get:

- int 1/u^2 du
=1/u + c
= secx + c

:. int secx(secx+tanx) dx = secx + tanx + c

And there you have it!