Continuous function f(t), t >= 0

[aselin0331]

Homework Statement

1+ integrand of e^t f(t) dt (from a to x^2) = ln (1+x^2)

Homework Equations

The Attempt at a Solution

I managed to sub in x^2 and derive the left then moved the one to the other side and differentiated the right hand side as well (a hint attached to the question: to find f(t) differentiate both sides in respect to x)

It's asking us to find a positive constant a. How do I do that?

Thanks :)

 

[hunt_mat]

Differentiate under the integral sign, there is a rule for this if
$$F(x)=\int_{a(x)}^{b(x)}f(x,t)dt$$
then:
$$F'(x)=f(x,b(x))b'(x)-f(x,a(x))a'(x)+\int_{a(x)}^{b(x)}\frac{\partial f}{\partial x}dt$$
Use this expression to find f(t), I obtain f(t) as:
$$f(t)=\frac{e^{-t}}{1+t}$$
Insert this into the integral and the value of a will pop out.