First slide

Introduction to integration

Question

$\begin{aligned} If \int g (x) d x = g (x), then \int g (x) \{f (x) + f^{'} (x)\} d x is e q u a l t o \end{aligned}$

Moderate

A

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

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

$g (x) f (x) + C$
D

$g (x) f^{2} (x) + C$

Solution

$\begin{array}{l} \int g (x) \{f (x) + f^{'} (x) ∣ d x \\ \begin{aligned} = & \{\int g (x) f (x) d x\} + \int g (x) f^{'} (x) d x i n t e g r a t i o n b y p a r t s \\ = & f (x) (\int g (x) d x) - \int (f^{'} (x) \int g (x) d x) d x + \int g (x) f^{'} (x) d x \\ = f (x) g (x) - \int g (x) f^{'} (x) d x + \int g (x) f^{'} (x) d x + C [∵ \int g (x) d x = g (x)] \\ = f (x) g (x) + C \end{aligned} \end{array}$

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