1. ## integral problmes

1. Find (f^-1)'(0) if:
integral sign x, 0 1+sin(sint)dt.
2. Find a function g such that:
integral sign x, 0 tg(t)dt= x+x^2
3. Find all continous functions f satisfying:
integral sign x,0 f=(f(x))^2+C. for some constant C.
4. Find F'(x) if F(x)= integral x,0 xf(t)dt. (The answer is not xf(x); perform obvious manipulatipon on the integral before trying to find F'.)
5. The limit lim integral sign N, a f, if it exists, is denoted by integral sign
N-> alpha
integral sign (alpha, a) f

a) Determine integral sign alpha, 1 x^r dx, if r<-1

2. Originally Posted by Swamifez
1. Find (f^-1)'(0) if:
integral sign x, 0 1+sin(sint)dt.
This function is not invertible .
2. Find a function g such that:
integral sign x, 0 tg(t)dt= x+x^2
Let, x>0
f(x)=INTEGRAL (from 0 to x) t*g(t) dt
We are told that,
f(x)=x+x^2

Take the derivative of both sides (apply FT of Calculus).
f'(x)=x*g(x)
f'(x)=1+2x
Thus,
1+2x=x*g(x)
Thus,
1/x+2=g(x)

3. Originally Posted by Swamifez
3. Find all continous functions f satisfying:
integral sign x,0 f=(f(x))^2+C. for some constant C.
I think it is more appropirate to say, "all differenciable functions".
---
INTEGRAL (from 0 to x) f(t)dt=(f(x))^2+C
Take derivative of both sides (apply FT of Calculus),
f(x)=2f'(x)f(x)
One solution is when,
f(x)=0 everywhere.
Otherwise we can divide by it (actually we cannot because there might be a zero point, but that will not happen because the function is differenciable)
1=2f'(x)
Thus,
1/2=f'(x)
Integrate,
1/2x+C=f(x)