Say I have a function,(adsbygoogle = window.adsbygoogle || []).push({});

f(x) = x sec (f(x)) [this is just an example function, the actual problem is more complicated]

g(x) = x f(x), then using integration by parts, I can write

I =_{a}∫^{b}g(x) dx =_{a}∫^{b}x f(x) dx = (f(x) [itex]\frac{x^{2}}{2}[/itex])|[itex]^{b}_{a}[/itex]- [itex]\frac{1}{2}[/itex]_{a}∫^{b}[itex]\frac{d f(x)}{dx}[/itex] x^{2}dx

Then can I write the integral as (cancelling dx from the numerator and denominator in integral on the RHS)?

I = (f(x) [itex]\frac{x^{2}}{2}[/itex])|[itex]^{b}_{a}[/itex]- [itex]\frac{1}{2}[/itex]_{f(a)}∫^{f(b)}x^{2}df

or,

I = (f(x) [itex]\frac{x^{2}}{2}[/itex])|[itex]^{b}_{a}[/itex]- [itex]\frac{1}{2}[/itex]_{f(a)}∫^{f(b)}f^{2}cos^{2}(f) df ?

This is simplified version of my actual problem. So any inputs will be very useful?

PS: I have edited the originally posted problem definition as it was causing confusion.

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# Is this a valid operation (integration by parts)?

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