Solve [itex]f'(x)*a(x)+a'(x)=1[/itex] For a(x)(adsbygoogle = window.adsbygoogle || []).push({});

(there's another less important question at the bottom)

Background behind equation (trying to find a function to integrate any e^f(x)):

[itex]\int e^{f(x)}\,dx=e^{f(x)}*a(x)[/itex]

[itex]e^{f(x)}=e^{f(x)}*{f'(x)}*a(x)+e^{f(x)}*a'(x)[/itex]

[itex]1=f'(x)*a(x)+a'(x)[/itex]

A few of my attempts:

[itex]df(x)*a(x)+da(x)=d(x)[/itex]

[itex]\int a(x)\,df(x)+a(x)=x[/itex]

[itex]a(x)=x-\int a(x)\,df(x)[/itex]

[itex]a(x)=x-\int x-\int x-\int x ...\,df(x)\,df(x)\,df(x)[/itex]

[itex]a(x)=x-\int x\,df(x)+\iint x\,d^2f(x)-\iiint x\,d^3f(x)...[/itex]

Note, this attempt only works if f(x) is an integrable function

This method doesn't work however, since [itex]\int e^x\,dx=e^x[/itex], You know the value of a and f(x) (1 and x respectively).

Substituting into the equation, the method stops working at line 2. Assuming that this is because you can only integrate a side of an equation with respect to a single function, I tried a different method.

[itex]f'(x)*a(x)=1-a'(x)[/itex]

[itex]\int f'(x)*a(x)\,dx=\int 1-a'(x)\,dx[/itex]

[itex]\int f'(x)*a(x)\,dx=x-a(x)[/itex]

A rearranged version of line two...

Where have I gone wrong in my attempts?

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# B Integrating any e^f(x) function

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