If [tex]f(0)=g(0)=0[/tex], show that(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int _0 ^a f(x) g ^{\prime \prime} (x) \: dx = f(a) g^{\prime} (a) - f^{\prime} (a) g (a) + \int _0 ^a f ^{\prime \prime} (x) g (x) \: dx[/tex]

I know I need to use integration by parts, but I'm having a hard time figuring out the right choice of [tex]u[/tex] and [tex]dv[/tex]. What I do know is the following:

[tex]\underbrace{\int _0 ^a f(x) g ^{\prime \prime} (x) \: dx}_{\int _0 ^a u \: dv} = \underbrace{f(a) g^{\prime} (a) - f^{\prime} (a) g (a)}_{\left. uv \right] _0 ^a} + \underbrace{\int _0 ^a f ^{\prime \prime} (x) g (x) \: dx}_{-\int _0 ^a v \: du}[/tex]

Also, the closest I get to [tex]\left. uv \right] _0 ^a[/tex] so far is

[tex]\frac{d}{dx} \left[ f(x) g(x) \right] = f(x) g^{\prime} (x) + f^{\prime} (x) g (x) \Longrightarrow \left. \frac{d}{dx} \left[ f(x) g(x) \right] \right|_0 ^a = f(a) g^{\prime} (a) + f^{\prime} (a) g (a)[/tex]

but, as you can see, that doesn't work because of a sign.

Any help is highly appreciated.

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# Homework Help: Integration by parts / proof

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