# Is there an error in my textbook?

Tags:
1. Oct 8, 2015

1. The problem statement, all variables and given/known data

2. Relevant equations

Doesn't the integral of sec x tan x equal sec x?

3. The attempt at a solution

2. Oct 8, 2015

### Ray Vickson

Your textbook is correct. The attachment of your work is too small and blurry for me to read, so I will not even try. I will look at it if you type it out.

3. Oct 8, 2015

Great, thanks :-) Isn't the integral of sec x tan x equal to sec x?

4. Oct 8, 2015

### Staff: Mentor

Yes, but I don't see how this fact applies to your question.

5. Oct 8, 2015

### Ray Vickson

Yes, as you can verify by taking the derivative. Likewise, you can check the book's answer by differentiation (which is something you should always do whenever you integrate).

6. Oct 8, 2015

### Daeho Ro

Well, it is correct, but the textbook uses the fact that $\frac{d\tan x}{dx} = \sec^2 x$. If you use $\frac{d(\sec x \tan x)}{dx} = \sec x$, then the first part of integration will be
$$\int \sec^2 x \tan^2 x dx = \int \sec x \tan^2 x d(\sec x \tan x),$$
which is not that easy to calculate. However, when you use $\frac{d\tan x}{dx} = \sec^2 x$, then
$$\int \sec^2 x \tan^2 x dx = \int \tan^2 x d(\tan x) = \dfrac{1}{3} \tan^3 x + C_1.$$

7. Oct 8, 2015