First slide

Methods of integration

Question

If $\int f (x) dx = f (x) + C, then \int {f (x)}^{2} dx$ is

Moderate

A

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

$\frac{1}{3} {f (x)}^{3} + C$
C

$f (x) + C$
D

$\frac{1}{2} {f (x)}^{2} + C$

Solution

Given,
$\begin{array}{l} \int f (x) dx = f (x) + C \\ f (x) = f^{'} (x) \\ ∴ \int \{f (x)^{2}\} dx = \int f (x) \cdot f^{'} (x) dx = \frac{1}{2} {f (x)}^{2} + C \end{array}$

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