# Finding the Indefinite Integral Extension Questions

1. May 21, 2012

### CaptainK

1. The problem statement, all variables and given/known data
∫8x3e-cos(x4+4)sin(x4+4)dx

2. Relevant equations
Let u = cos(x4+4)

3. The attempt at a solution
I know the answer does not have the sin in it and only the e remains, because when the integral is found e stays unchanged.
I could find somewhere online to calculate it, but I want to know how to do it, because questions like this will be on the final exam.

2. May 21, 2012

### Dick

You have the correct u substitution. Now use it. What's du?

3. May 21, 2012

### micromass

So, what do you get after that u substitution??

4. May 21, 2012

### CaptainK

So I've found du and put it into the form

∫udv = uv - ∫vdu

so for v I have 8x3

du = -sin(4x3) dx

dv = 24x2 dx

Which gives me

∫cos(x4+4)24x2dx = cos(x4+4)8x3 - ∫8x3(-sin4x3) dx

But I feel like I'm going down the wrong path, especially since the e isn't present

5. May 21, 2012

### Dick

You are going down the integration by parts path. It's not the right way to go. Why do you think du=(-sin(4x^3))dx?

6. May 22, 2012

### CaptainK

So using substitution

du = sin(x4+4)

because cos converted to sin doesn't change its sign, I thought it did but its for converting from cos to sin.

t = u

∫f'(g(x))g'(x) dx = ∫f'(t) dt/dx = ∫ f'(t)dt = f(t) + C

= f(g(x)) + C when substituting back in

f(x) = original equation and
t = u = cos(x4+4)

7. May 22, 2012

### Dick

df(x)=f'(x)*dx. f(x) here is cos(x^4+4). What is df(x)?

8. May 22, 2012

### CaptainK

so f(x) = cos(x4+4)

then

f'(x) = sin(x4+4) + 4x3

which gives

∫cos(x4+4) +4x3 *sin(x4+4) + C

9. May 22, 2012

### Dick

I'd suggest you review the chain rule. If f(x)=cos(x^4+4) then the derivative f'(x) is not what you wrote. Give it another try. Tell me what the chain rule says.

Last edited: May 22, 2012
10. May 22, 2012

### CaptainK

Chain Rule states

dy/dx = dy/du * dx/du

∫8x3e-cos(x4+4)sin(x4+4)

8∫x3∫e-cos(x4+4)∫sin(x4+4)

8∫x4/4∫e-cos(x4+4)∫cos(x4+4)+4x3

11. May 22, 2012

### Dick

Stop worrying about the integral for a while. You are having trouble differentiating f(x)=cos(x^4+4). Try and get that right first.

12. May 22, 2012

### CaptainK

Finding the derivative of cos(x4+4)

I got -4x3sin(x4+4)

Do I then follow the formula above with f(x) and t and then solve for the Integral.

13. May 22, 2012

### Dick

Ok, you've got the derivative. If you take u=cos(x^4+4), that means du=(-4x^3)sin(x^4+4)dx. Now you should be able to turn the integral into
$$C \int e^{-u}du$$
where C is a numerical constant. This is u-substitution.