# How Do You Integrate tan(x)sec^2(x)?

• scorpa
In summary, I am having trouble integrating tanxsec^2x. I tried doing something with the chain rule but it didn't work out at all. I know that I am supposed to show my work, but at this point I don't have anything to show you guys. Thanks a lot for any help you can give me.
scorpa
Hello everyone,

I am having some trouble finding the integral of tanxsec^2x. I honestly have no idea where to go with this one. I've finished all the others but this one is really screwing me up. I tried doing something with the chain rule but it didn't work out at all. I know that I am supposed to show my work, but at this point I don't have anything to show you guys. Thanks a lot for any help you can give me.

scorpa said:
Hello everyone,

I am having some trouble finding the integral of tanxsec^2x. I honestly have no idea where to go with this one. I've finished all the others but this one is really screwing me up. I tried doing something with the chain rule but it didn't work out at all. I know that I am supposed to show my work, but at this point I don't have anything to show you guys. Thanks a lot for any help you can give me.

Write everything in terms of sines and cosines. Then you can do a simple substitution.

Or just note that the derivative of tangent is the secant squared... u-substitution, anybody?

--J

Ok guys, thanks for the help, but I'm still completely lost. I still don't have a clue of what I should do.

Are you familiar with u-substitution? You're going to have to make one to evaluate the integral. You have two choices for what to let u equal, both work. What do you think u might equal?

--J

I am sort of familiar with it. Could you let u equal tanx?

scorpa said:
I am sort of familiar with it. Could you let u equal tanx?

If you did, what would du be?

Ok, actually I might let u = sec^2x, then du should equal tanx+c?

scorpa said:
Ok, actually I might let u = sec^2x, then du should equal tanx+c?

That is not correct

You should review the derivatives of all the trig functions.

You're integrating u to get du for some reason. You must differentiate u to get du.

--J

$$\int \sec^{2} x \tan x \ dx=-\int \frac{d {}\cos x}{\cos^{3} x}=\frac{1}{2}\sec^{2}x +\mathcal{C}$$

Daniel.

shouldnt it be:
$$\int sec^2x~tanx~dx=\int\frac{sinx}{cos^3x}dx$$
then, you could do u-sub and set $u=cosx$ and go from there?

*Edit*
dextercioby, you put sinx instead of cosx. i think tanx = sinx/cosx
*Edit*

Last edited:
p53ud0 dr34m5 said:
shouldnt it be:
$$\int sec^2x~tanx~dx=\int\frac{sinx}{cos^3x}dx$$
then, you could do u-sub and set $u=cosx$ and go from there?

and du would be

$$du = d cosx = -sinx dx$$

giving

$$sinx dx = -d cosx$$

which is what Dexter did to get his result, and what earlier hints were pointing to. It could also be done using

$$u= tanx$$

$$du= d tanx = sec^2x dx$$

$$sec^2x dx = d tanx$$

as suggested by Justin early on

Oh ok...I think I am starting to get it

## What is an integral of a trigonometric function?

An integral of a trigonometric function is a mathematical technique used to find the area under a curve of a trigonometric function. It involves finding the antiderivative of the function and evaluating it at the upper and lower limits of integration.

## What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. These functions relate the angles of a right triangle to the lengths of its sides.

## How do you integrate a sine function?

To integrate a sine function, you can use the formula ∫sin(x)dx = -cos(x) + C, where C is the constant of integration. This formula can be derived using the power rule for integration and the derivative of cosine.

## Can the Pythagorean identities be used to integrate trigonometric functions?

Yes, the Pythagorean identities, such as sin^2(x) + cos^2(x) = 1, can be used to simplify integrals of trigonometric functions. They can also be used to convert between different trigonometric functions, making integration easier.

## What are some common applications of integrals of trigonometric functions?

Integrals of trigonometric functions are commonly used in physics, engineering, and other fields to calculate the work done by a force, the displacement of an object, or the motion of a particle. They are also used in calculus to find the volume of irregular shapes and the length of curves.

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