Integration - The Basics

Hi,

I understand how to get to the answer, that's fine. But I just don't understand the logic used. "Consider", what do they mean? Are we just "Letting" y=... or is y=sin(2x+3) the actual integral?

If so, if we differentiate this value, we should get back to cos(2x+3) but we clearly don't. So this value cannot be the integral? Why are we taking these steps?

Thanks!

The answer is y=1/2 sin(2x+3)

If you differentiate that you get cos(2x+3)

The text is explaining why you you need the 1/2

I want to know whether step 1 is just a "rough" integration


Posted from TSR Mobile

No

Step 1 is telling you the thinking process

And what is the thinking process? An estimation?


Posted from TSR Mobile

I want to know whether step 1 is just a "rough" integration


Posted from TSR Mobile

It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function


I am struggling to see what is confusing you

Step 1 is an "intelligent " guess. You then check whether it is correct and adjust if necessary.
You should try to get to know certain standard patterns. For example, consider a function of x f(x). If the integral is F(x) then the integral of f(ax) will be F(ax)/a
Similarly the integral of f(ax+b) will be F(ax+b)/a.
IN reverse, if the derivative of f(x) is g(x) say then the derivative of f(ax) will be ag(ax) and of f(ax+b) will be ag(ax+b).
More generally the derivative of f(g(x)) will be g'(x)f'(g(x))
See page 38 in the attachment.

Thank you soooo much, I understand it now


Posted from TSR Mobile

You''re welcome

It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function


I am struggling to see what is confusing you

You''re welcome

Because say if I have the Q was integrate 1/(2x+1)^2 and I followed the 3 step method, I would get:


But the actual answer is -1/2(2x+1)^2


Posted from TSR Mobile

Both this question and the original question use the idea of "reverse chain rule"

You should know that differentiating the function of a function needs the chain rule

So if you are integrating the function of a function you will consider reversing the chain rule

I have used the reverse chain rules


Posted from TSR Mobile

Sorry did not look at your work properly

You need to increase the power

You have changed a power of -2 to a power of -3

That is decreasing

Sorry did not look at your work properly

You need to increase the power

You have changed a power of -2 to a power of -3

That is decreasing

Or do I have to account for the reciprocal?


Posted from TSR Mobile

I can see that you have changed from a 2to a 3

As you have realised now ... You have changed -2 to -3 ... Decreasing rather than increasing ... Since the function is in the denominator