Finding an antiderivative

[kggirl]

Can someone please tell me if this is correct:

Find an antiderivative for cos[2πx]

This is the answer that I got: sin π^2 + C[/list]

 

[jsbeckton]

If this is the problem: \(\displaystyle \int {\cos (2nx)dx}\) Then the antiderivitive should be \(\displaystyle - \frac{1}{{2n}}\sin (2nx)+c\)
You have to take the integral of the function and divide by cooeficient of the inner function.

 

[tkhunny]

I'm relatively positive that is "two-pi-x", but it makes NO difference in the process.

kggirl's answer is absent the original variable. How can that be right, kggirl?

 

[kggirl]

evaluate the integral using substitution

Evaluate Integral from 0 to 1 for x/(x^2-5)^3

Is this correct:

the derivative of x/(x^2-5)^3 is x/(2x-0)^3 = x/(2x)^3

so substitute 1 in for x: 1/(2*1)^3 = 1/8

 

[jsbeckton]

If you are evaluating the integral, you do not need to take the derivitive, the integral is sometimes refered to as the antiderivitive, it "undoes" the derivitive. Try again, just take the integral, use a U substitution if you need to.

let U = \(\displaystyle x^2 - 5\)
so du=2x dx

Now the integral becomes \(\displaystyle \frac{1}{2}\int\limits_{ - 5}^{ - 4} {u^{ - 3} du}\)


You have a "x" in the numerator but you need a "2x" so you have to multiply the integral by 1/2 to make up for that. When you use a U sub, you need to change to bounds by plugging in the old bounds to the u= equation. Is this ringing any bells?

 

[kggirl]

I don't understand why the integral from 0 to 1 became 5 to -4

 

[jsbeckton]

put the original values 0 & 1 throught the u equation

u=\(\displaystyle x^2-5\)

u=\(\displaystyle 1^2-5\)
=-4

u=\(\displaystyle 0^2-5\)
=-5

 

[jsbeckton]

note: You do not have to change the limits of integration, if you want you can change every u back to \(\displaystyle x^2-5\) after you integrate. Personally, I think that its easier to change the bounds of integration as I suggested.

 

[stapel]

Personally, I think that its easier to change the bounds of integration as I suggested.

But it is a matter of preference. For instance, I would rather switch everything back to x's and leave the limits alone. However, if you do this, you must switch from the definite integral (with limits) to the related indefinite integral (without limits, but with a constant of integration) if you want your work to be counted as correct. You can't just ignore the limits or the fact that, in the definite integral, they will have changed when you went from the x-based integral to the u-based one.

Eliz.

 

[jsbeckton]

I guess I always change them because I've been caught using the old limits with the u sub too many times. I think there is less chance for error if you change them and then, you usually have a much more simple computation at the end since you have made to equation easier with the u sub.

But like Stapel said, its a personal preference and at the end you should arive at the same answer. I would only suggest that you pick one method and STICK WITH IT!