I really don't get the substitution rule

[tony873004]

I really don't get the substitution rule. This is supposed to be the easiest problem in the homework set: u=3x
$$\int {\cos \,3x\,\,dx\,\, = \,\,\int {\cos \,u\,\, = \,\,\sin \,u + C\,\, = \,\,\sin 3x + C} }$$

But the right answer is 1/3 sin(3x). Where did the 1/3 come from?

 

[drpizza]

Look at your second integral in here:
$$\int {\cos \,3x\,\,dx\,\, = \,\,\int {\cos \,u\,\,dx?\,\, = \,\,\sin \,u + C\,\, = \,\,\sin 3x + C} }$$

with respect to what variable? x?
or, did you mean to have a du in there?

If you meant for a du to be there, what makes you think that du can be substituted for dx? You wrote u=3x. if I take the derivative, with respect to x, I get du/dx = 3. Multiplying both sides by dx (thanks, Leibnitz for that notation), I get du = 3 dx. And from that, dx will be equal to 1/3 du

 

[drpizza]

Actually, I think a "harder" integral will make this easier to see:
$$\int {\sec x \sec x \tan xdx$$
Let u = secx
du/dx = secxtanx
du = secxtanxdx

Now, when you substitute, u will take care of the first secx.
But, du will take care of the rest of it: secxtanxdx

 

[tony873004]

I think I did it right, but I still have a question.

$$u = 3x,\,\,\,\,\frac{{du}}{{dx}} = \left( {3x} \right)^\prime = 3,\,\,\,\,dx = \frac{{du}}{\begin{array}{l} 3 \\ \\ \end{array}},\,\,\,\,\int {\cos \,3x\,\,dx\,\,} = \,\,\int {\cos \,u\,\,\frac{{du}}{3} = \frac{{\sin u}}{3}} = \frac{{\sin 3x}}{3}$$

So when I integrate, the du simply disappears?

I hate Leibnitz notation. I do not think it is intuitive. I wish in Calc 1 they would have spent a lecture on Leibnitz notation. Rather, they just started using it without describing it.

 

[mattmns]

Yes, the du disappears with the integral sign (as would dx, dy, etc). Kind of like: 4 + 3 = 7, we don't leave the + there.

 

[z-component]

$$\int {\cos \,u\,\,\frac{{du}}{3}$$

At this point you should factor out the constant, 1/3, from the integral:

$$\frac{1}{3} \int {\cos \,u\,\,du$$

 

[tony873004]

...Kind of like: 4 + 3 = 7, we don't leave the + there.

:rofl: Thanks to you, I will never forget that.

Thanks, everyone. This was so confusing to me until I asked here. I just plowed through the next 10 problems with ease. But I'm stuck on the 11th, so I'm giving up for the night.

 

[jermanie]

Man, that was an easy problem. Think forward of the answer. I guess light years will come up for your mind to catch up.

 

[cristo]

Man, that was an easy problem.

It's only easy if you know the answer! When learning, any problem can be difficult if you can't see what to do!