First slide

Introduction to integration

Question

$\int [f (x) g^{''} (x) - f^{''} (x) g (x)] d x is equal to$

Moderate

A

$\frac{f (x)}{g^{'} (x)}$
B

$f^{'} (x) g (x) - f (x) g^{'} (x)$
C

$f (x) g^{'} (x) - f^{'} (x) g (x)$
D

$f (x) g^{'} (x) + f^{'} (x) g (x)$

Solution

$\begin{array}{l} \int [f (x) g^{''} (x) - f^{''} g (x)] dx \\ = \int f (x) g^{''} (x) dx - \int f^{''} (x) g (x) dx \\ = (f (x) g^{'} (x) - \int f^{'} (x) g^{'} (x) dx) \\ - (g (x) f^{'} (x) - \int g^{'} (x) f^{'} (x) dx) \\ = f (x) g^{'} (x) - f^{'} (x) g (x) \end{array}$

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