[Math] Integrate $\int\sec ^2 x \tan x\, dx$

calculusintegration

I am trying to compute
$$
\int\sec^2 x \tan x\, dx.
$$
I substituted $u =\sec^2 x$ to get the integral as
$$
\frac{\sec^4x}{2}
$$
as my answer, but according to the textbook I am using, I'm wrong. Can anyone help me with the correct answer? According to my textbook, the correct option is between (a) $2\sec^2 x$ (b) $\tan x$ (c) $1/2 \sec x$ or (d) $\csc x \cot x$

Best Answer

When you do a substitution like that in an integral, you want to end up with something that looks like this:

$$\int u \; du.$$

If $u = \sec^2 x$, what is $du$? If it is not $\tan x\;dx$ then you will not be able to change $\sec^2 x \tan x \; dx$ to $u \; du$.

Try a different substitution. There are not many choices, in fact I see only three likely possibilities and $u = \sec^2 x$ is the only one of the three that does not work.

EDIT: John Joy, in another answer, shows that the substitution $u = \sec^2 x$ actually does work if you do it correctly. You do not get anything in the form of $\int u\;du$ that way; instead, you get something easier. I should have acknowledged earlier that while $\int u\;du$ is one form you might hope to achieve from a substitution, really the point is just to get the integral into some form you know how to solve.

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