Integrals of differentialble functions on a bounded interval

[mjpam]

Given two functions $f_{1}(x)$ and $f_{2}(x)$ that are differentiable on $[a_{1},a_{2}]$ and $\int^{a_2}_{a_1}f_1(x)f'_{2}(x)dx=b_2$, how would one calculate $\int^{a_2}_{a_1}f_1(x)f'_{2}(x)dx$?

This is not a homework problem. I saw it on the internet and realized that I did not know where to begin solving it.

 

[HallsofIvy]

I am confused. Are you asking how to find $b_2$? If so, that would depend strongly on what $f_1$ and $f_2$ are. There are many different ways to evaluate various integrals, and, in fact, if we take $f_x(x)= 1$, for all x, and $f_2(x)= e^{-x^2}$, both differentiable functions, then it is easy to show that
$$\int_{a_1}^{a_2}f_1(x)f_2(x)dx= \int_{a_1}^{a_2}e^{-x^2}dx$$
exists but there is no analytic method of finding the integral. The best you could do is use some numeric method, such as Simpson's rule, to find the value.

 

[micromass]

Are you perhaps asking how to calculate $$\int_{a_1}^{a_2}{f_1^\prime(x)f_2(x)dx}$$?? This can be easily done by "integration by parts".

The answer is $$\int_{a_1}^{a_2}{f_1^\prime(x)f_2(x)dx}=f_1(a_2)f_2(a_2)-f_1(a_1)f_1(a_2)-b_2$$.

 

[mjpam]

Are you perhaps asking how to calculate $\int_{a_1}^{a_2}{f_1^\prime(x)f_2(x)dx}$?? This can be easily done by "integration by parts".

The answer is $\int_{a_1}^{a_2}{f_1^\prime(x)f_2(x)dx}=f_1(a_2)f_2(a_2)-f_1(a_1)f_1(a_2)-b_2$.

I think that's correct.

My question was:

Given two functions $f_{1}(x)$ and $f_{2}(x)$ differentiable on a closed interval $[a_{1},a_{2}]$, two constants $c_{1}$ and $c_{2}$, and the facts that $\int_{a_{1}}^{a_{2}}f_{1}f_{2}^{\prime}dx=c_{1}$ and $\int_{a_{1}}^{a_{2}}f_{1}^{\prime}f_{2}dx=c_{2}$, can $c_{2}$ be expressed in terms of $c_{1}$?