Heeelp with this antiderivative (silly question prob.)

[frasifrasi]

As an example for the tabular method, my book says that the antiderivative of

sin 4x is -1/4 cos 4x...

My calc I fundamentals are obviously rusty, so can anyone explain why it is -1/4cos 4x as opposed to just cos 4x like i thought?

thank you!

 

[rocomath]

differentiate your "opposed" and tell me what you get

what steps would you take in differentiating your answer?

what is the derivative of cos? will it be positive or negative? if you have an angle with constants on it or in general, is not just x, what would you have to do in differentiating?

 

[PowerIso]

You can use a u sub to show why it would be -1/4

The Integrand of Sin(4x)dx

u=4x du = 4dx du/4=x

So 1/4Integrand of Sinu du

Now Integrate.

The Antidev of Sin(u) is -Cos(u)

Now that is being multiplied by 1/4 too

So you get -1/4 * Cos(4x)

 

[frasifrasi]

Thank you poweriso, you actually helped. Correct me if i am wrong, but when you get something like the integral of sin(4x), isn't standard to assume it is -cos (4x) like 4x = x?

It is the first time I have seen this on the text.

 

[TimNguyen]

Thank you poweriso, you actually helped. Correct me if i am wrong, but when you get something like the integral of sin(4x), isn't standard to assume it is -cos (4x) like 4x = x?

It is the first time I have seen this on the text.

I think the general form would aid you.

Int(sin(kx),x) = -1/k cos(kx) + C
Int(cos(kx),x) = 1/k sin(kx) + C

where k is a constant

Edit: Never mind, you're using substitution.

For the problem you stated, you could replace the "4x" with "u", but you must also change the "dx" to some form of "du". In your case, if "4x" = "u", then "4dx" = "du" by differentiating both sides. Then you could see that you could substitute 4x with you and dx with du/4. From there, you would have:

Int(sin(u) du/4) = 1/4 Int(sin(u) du) = -1/4 cos(u) + C.

Then you substitute back in 4x for u, which gives:

-1/4 cos(4x) + C

 

[turdferguson]

Is the derivative of sin(4x) equal to cos(4x)? The chain rule has an integration counterpart

 

[PowerIso]

Thank you poweriso, you actually helped. Correct me if i am wrong, but when you get something like the integral of sin(4x), isn't standard to assume it is -cos (4x) like 4x = x?

It is the first time I have seen this on the text.

No and you can tell it's wrong by doing the derivative. You'll have to use the Chain rule because your x is actually an x times a constant which makes it a more complicated x.

So -Cos(4x) would be Sin(4x) * 4

So you do not get your F(x). It is pretty obvious that they differ by a constant though.

So like it has been stated before, generally speaking, when you have constant * x within a trig function it'll turn into 1/k where k is a constant * trig function.

I wouldn't worry to much about that though. If you can do U sub you can figure it out easily enough. Later on, it'll be as naturally as adding numbers in your head.