Differential Equation Integral.

[PFStudent]

1. Homework Statement .
Let,
$${{f(x)}, {{{f}^{\prime}}{\left(x\right)}}, {{{f}^{\prime\prime}}{\left(x\right)}},...,} = {{f}, {{f}^{\prime}}, {{f}^{\prime\prime}},...,}$$

Prove (without just differentiating the RHS),
$${{\int_{}^{}}{{f}^{\prime}}{\left({{f}+{{f}^{\prime\prime}}}\right)}{dx}} = { {{\frac{1}{2}}\left({f}^{2}}+{{{f}^{\prime}}^{2}\right)}+{C} }$$

2. Homework Equations .
Knowledge of Calculus and Differential Equations.

3. The Attempt at a Solution .
In the lecture notes the above problem was presented as part of another proof. I'm really not sure where to begin on this. Maybe integration by parts?

Thanks,

-PFStudent
EDIT: Thanks Mark44 for the edit.

 

[rock.freak667]

Expand out the integrand, then split it into two integrals.

For example

$$\int x(x+1) dx = \int (x^2+x)dx= \int x^2 dx + \int x dx$$

 

[Mark44]

1. Homework Statement .
Let,
$${{f(x)}, {{{f}^{\prime}}{\left(x\right)}}, {{{f}^{\prime\prime}}{\left(x\right)}},...,} = {{f}, {{f}^{\prime}}, {{f}^{\prime\prime}},...,}$$

Prove (without just differentiating the RHS),
$${{\int_{}^{}}{{f}^{\prime}}{\left({{f}+{{f}^{\prime\prime}}}\right)}{dx}} = {{{f}^{2}}+{{{f}^{\prime}}^{2}}+{C}}$$

2. Homework Equations .
Knowledge of Calculus and Differential Equations.

3. The Attempt at a Solution .
In the lecture notes the above problem was presented as part of another proof. I'm really not sure where to begin on this. Maybe integration by parts?

Thanks,

-PFStudent

You're missing some factors on the RHS. It should be 1/2 [f']^2 + 1/2 [f'']^2 + C.

You don't need integration by parts; an ordinary substitution will do for each integral. integration with