Mastering Integrals: Solving for cos(2x) and Other Trigonometric Functions

[sunfleck]

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

 

[Dick]

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

 

[sunfleck]

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--

 

[Dick]

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.

 

[sunfleck]

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

 

[Mark44]

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