# Integral of sec²(x)|Two ways

Hi everyone! I know it’s been a really long time since my last post, but lessons during COVID-19 really had me busy and stressed out… but here I am again! Hope you are all doing great and ready for more integrals!

In this post I am going to show you two methods I found to integrate the function *y*=sec²*x*. The first one requires a bit of knowledge about trigonometry and trigonometric identities and might not be very intuitive, whereas the second involves a substitution right at the beginning and then some algebra and trigonometric identities. Feel free to share more methods if you know them!

**First method**

First of all, what is sec(*x*) equal to? It’s simply the reciprocal of the function cos(*x*), i.e. 1/cos(*x*). Therefore, secant squared is equal to one over cosine squared. That simple!

Now comes the not very intuitive part. In trigonometry, 1 is given by the sum of the squares of the two fundamental functions sine and cosine. This is the most important trigonometric identity and the one we are going to use:

At this point we basically have

It’s like (*a*+*b*)*(1/c)=*a*/*c*+*b*/*c*: we are simply distributing the 1/*c*.

Let’s apply linearity:

Remember the property

where *m* is a constant, in our case 1; therefore

The problem here is mostly the fact that we have sine squared at the numerator. If it was simply sin(*x*), we would have performed a *u*-substitution by letting *u*=cos(*x*) and its derivative, -sin(*x*), would have gone to the denominator and so it would have cancelled out with the sin(*x*) on top.

Notice that cos²(*x*) becomes *u*² because we let *u*=cos(*x*).

Let’s apply the power rule for integrals. Just in case you don’t remember,

Let’s undo the substitution (remember: *u*=cos(*x*))

BUT we don’t have this integral; instead we have

But after all, sin²(*x*) is sin(*x*)*sin(*x*), so:

We have a product, which means we can perform integration by parts, and naturally we are going to differentiate sin(*x*) and integrate the other term, since we have already found the antiderivative (1/cos(*x*));

Remember that

Therefore,

This is the result of

but we also have that *x* from the first part of the integration:

sin(*x*)/cos(*x*) is tan(*x*); finally,

**Second method**

This time we are going to let *u* be sec(*x*), so that sec²(*x*) becomes *u*²:

sec(*x*) is 1/cos(*x*), so in order to differentiate it we apply the reciprocal rule:

Therefore,

Now, there is a problem: if we substitute we must integrate with respect to the new variable, *u* in our case, which means every function of the “old” variable *x* must be written in terms of *u*. We have

The terms in red are simply *u*²:

We’re almost there. How can we write sin(*x*) using cos²(*x*)? Remember the identity sin²(*x*)+cos²(*x*)=1:

We know that *u*²=1/cos²(*x*), therefore cos²(*x*)=1/*u*²:

This means

Let’s give a look at the integral:

We want d*x*, therefore

This is exactly what we wanted, no more *x*‘s. At this point what do you think we can do? Give it a try and then keep reading to see if you got it right.

This is what we are going to do: if you differentiate *u*²-1, you’ll get 2*u*, which will be the d*u* but, since we want d*x*, it will “become” 1/2*u* and so it will cancel out with the *u* at the numerator. Since we used *u* before, we’ll change variable.

1/2 is a constant and is moved in front of the integral; at this point we can apply the power rule for integrals:

We are almost done; let’s undo all substitutions. Remember:

That’s the “final” result; “final” because we can actually do more: sec²(x)-1 comes from the trigonometric identity sec²(x)-tan²(x)=1 and, as you can see, it is equal to tan²(x); this means that

Finally,

That’it! We’ve integrated the function sec²(*x*) with two methods. If you weren’t able to do it by yourself, keep in mind that looking at the procedures is actually a great way to learn the mechanism and all of the techniques; this is how I started, so stay positive!

I really hope you found this post helpful and if you enjoyed it leave a like and subscribe for more! In the next post we’ll find the integral of csc²(*x*), so in the mean time, give it a try!

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