Math Help - Integral problems

1. Integral problems

Hey guys, it's my first time here! So i'm not sure if this is considered pre-cal or cal ( because where I'm from cal 1 and 2 isn't taught in uni so it might be pre :P). Also I'm sorry for my integral notation, first time I write the notation online!

I have few problems if you guys don't mind pointing me in the right directions ( final tomorrow zzz):
1) Find all the values of x such that

int (t^3-t) dt from 0 to x = (1/3) int(t-t^3) from sqrt(2) to x

I tried to just integrate both side and equate them then just solve for x but I don't seem to get the right answer; is integrating both even the right way to go?

2) Let f(x)= int( ln(t^2 + 1) dt from 1+sinx to x. Find f'(0)

Does this means I have to use the fondamental theorem of calculus 1? I've been trying to solve this but it seems impossible. It's look so so simple though!

3) Find the minimum value of int(sqrt(t)) dt from x to x^2

I don't even know where to start this one :/

Thanks to anybody who can point me in the right direction. I don't need the work told to me, I just want pointers. I know my stuff I just can't get my head around those problems.

2. Re: Integral problems

Hey MarcLeclair.

Can you show us youatr working for 1?

For 2) Yes you need to use FTC but remember that you have a 1 + sin(x) and not an x so you will need to use the chain rule.

For 3) try considering that you have a function given by the integral and to find a maximum, you find when the derivative of that function is 0 with 2nd derivative being less than zero.

3. Re: Integral problems

yeah of course,

for one I did

t^4/4 - t^2/2 from 0 to x = (1/3) (t^2/2 - t^4/4) (sqrt(2) to x)
= (1/3) ( (x^2/2 - x^4/4) - ( sqrt(2)^2/2 - sqrt(2)^4 /2) ) the last bracket gives 0 I think?
(x^4/4 - x^2/2 - x^2/6 +x^4/12 = 0

And then I can't simplify that much x.x

4. Re: Integral problems

Wait for 3.... If I find f(x) ( being the integral) than find the derivative I come back to the same thing :/

5. Re: Integral problems

So you have (x^4/4 - x^2/2) - 1/3 * [(x^2/2 - x^4/4) - (1/2 - 1)] = 0

Since you have the form ax^4 + bx^2 + c = 0, try factorizing and use u = x^2 to get

au^2 + bu + c = 0 and use the quadratic formula to get u and then plug u = x^2 to get x.

6. Re: Integral problems

aaaaah. I didn't see I could substitute it there. So simple .... thanks a lot chiro^^