MI
Answered byMolly I.Maths Tutor

Show that the integral of tan(x) is ln|sec(x)| + C where C is a constant.

First, recall that tan(x) can be rewritten in terms of sine and cosine.

tan(x) = sin(x)/cos(x)

The rephrasing of our question suggests that we should try the substitution rule of integration.

We should substitute u=cos(x), since then du = -sin(x) dx and so sin(x) dx = -du

So the integral of tan(x) = the integral of sin(x)/cos(x) = the integral of -1/u = - ln|u| +C = - ln|cosx| +C

Now, - ln|cos(x)| = ln(|cos(x)|-1) = ln(1/|cos(x)|) = ln|sec(x)|

Therefore, the integral of tan(x) is ln|sec(x)| + C

Related Maths A Level answers

All answers ▸

Integrate x/(x^2+2)


What is the chain rule?


Consider the unit hyperbola, whose equation is given by x^2 - y^2 = 1. We denote the origin, (0, 0) by O. Choose any point P on the curve, and label its reflection in the x axis P'. Show that the line OP and the tangent line to P' meet at a right angle.


What is the moment about the pivot C


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning