Is My Interpretation of the Integral Rule Correct?

[christian0710]

Hi I'm tying to understand notations in Integration, and would really appreciate some help making sure that my understanding is right.My books writes

Let u and v be functions of x whose domains are an open interval I, and suppose du and dv exist for every x in I.Then it defines

1) ∫(du)= u + C
2) ∫(c*du) = c*∫(du)and
3) ∫(cos(u)du = sin u +CNow i do understand the first 2, but I want to make sure i understand the 3rd rule.

If u is a function of x with the equation u(x)=x^2

Then the derivative
du/dx= 2x

The differential
du=u'(x)dx

Now if it's true that du=u'(x)dx
Then it does make sense that ∫du =u+C because ∫du=∫u'(x)*dx and the integral of the derivative if u is u.

But if u=x^2

Then ∫(cos(x^2)*du = ∫(cos(x^2)*(2x)dx and this is = sin(x^2) + C as the statement above says.
because the derivative of sin(x^2) = cos(x^2)*(2x).

Is this the right interpretation?

 

[cpsinkule]

why is ∫(cos(x^2)*(2x)dx not equal to sin(x^2) + C? you said in the line below it that

derivative of sin(x^2) = cos(x^2)*(2x).

which is exactly what appears under the integral sign.

 

[christian0710]

why is ∫(cos(x^2)*(2x)dx not equal to sin(x^2) + C? you said in the line below it that

which is exactly what appears under the integral sign.

Oh forgive me, that NOT was a BIG mistake ;)
You are right, it is e equal to it.