# Integrals: cos(2x)

Hello :)

I'm not sure how to deal with an integral of the form cos(2x), sin(2x), tan(3x), etc. Its the constant that is throwing me off - I know what the antideriv of cosx, sinx or tanx is. My text doesn't seem to explain what to do with the constant and I can't seem to find the info using a direct internet search so I was wondering if someone would please tell me what to do with the constant?

Help would be greatly appreciated

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Dick
Homework Helper
Substitute u=2x, du=2*dx. Do the u integral and substitute back.

Substitute u=2x, du=2*dx. Do the u integral and substitute back.

Thanks, but we actually haven't covered the sub rule yet so I'm wondering if maybe there is a different way to get the answer?

Updating because I found a similar question in my text...
The text has a problem showing..

$$\int sin( \Pi x)dx = -(1/ \Pi )cos( \Pi x)dx$$

So I take it if f(x) = cos(2x)dx then F(x) = (1/2)sin(2x)... but the chpt explaining the sub rule says something about changing the boundaries of a definate integral when using the sub rule.... Would I have to do anything to the boundaries? (I've been asked to solve using FTC where $$\int \stackrel{b}{a} = F(b) - F(a)$$ ) --I found that problem in the sub rule section which is why I'm confused about the boundaries--

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Dick
Homework Helper
Ok. The integral of sin(2x) is -(1/2)*cos(2x). You can check that just be differentiating -(1/2)*cos(2x) and seeing that you get sin(2x). The antiderivative -(1/2)*cos(2x) is expressed in terms of the original variable x. So the limits are the same as the original limits. You don't have to change them.

Oh right.. that makes sense! Thanks for taking the time to reply :)

Mark44
Mentor
Thanks, but we actually haven't covered the sub rule yet so I'm wondering if maybe there is a different way to get the answer?
I don't think there's a substitution rule, per se. Substitution is nothing more than applying the chain rule of differentiation to antidifferentiation.
Updating because I found a similar question in my text...
The text has a problem showing..

$$\int sin( \Pi x)dx = -(1/ \Pi )cos( \Pi x)dx$$

So I take it if f(x) = cos(2x)dx then F(x) = (1/2)sin(2x)... but the chpt explaining the sub rule says something about changing the boundaries of a definate integral when using the sub rule.... Would I have to do anything to the boundaries? (I've been asked to solve using FTC where $$\int \stackrel{b}{a} = F(b) - F(a)$$ ) --I found that problem in the sub rule section which is why I'm confused about the boundaries--
To expand on what I said above, if g(x) = sin(2x), then g'(x) = 2cos(2x). Looking at things from the opposite direction,
$$\int 2cos(2x) dx = sin(2x) + C$$
Equivalently,
$$\int cos(2x) dx = 1/2 sin(2x) + C_1$$
where C1 = C/2.

Here's a slightly more complicated problem:
$$\int xcos(x^2) dx$$

For this problem, I can see that part of the integrand is cos(x^2), so I think of this as being cos(u(x)), where u(x) = x^2. I see also that du/dx = 2x, so it's helpful that the other factor in the integrand is x.

So x*cos(x^2)*dx = 1/2 * du/dx * cos(u) * dx = 1/2 * cos(u)*du

I know that
$$\int cos(u) du = sin(u) + C_1$$
so

$$\int 1/2*cos(u) du = 1/2*sin(u) + C$$

= 1/2*sin(x^2) + C