First slide

Methods of integration

Question

The function $f$ whose graph passes through $(4, - 20)$ and whose derivative is $\cos (\sqrt{4 - x})$ is given by

Moderate

A

$\sqrt{4 - x} \sin \sqrt{4 - x} + (4 - 6 x) \cos \sqrt{4 - x}$
B

$\sqrt{4 - x} \sin \sqrt{4 - x} + \sqrt{4 - x} \cos \sqrt{4 - x} - 20$
C

$- 2 (\sqrt{4 - x} \sin \sqrt{4 - x} + \cos \sqrt{4 - x}) - 18$
D

$- (2 x + 12) \sqrt{4 - x} \sin \sqrt{4 - x}$
$+ (4 - 6 x) \cos \sqrt{4 - x}$

Solution

$\begin{aligned} f (x) = \int \cos \sqrt{4 - x} d x + C \\ = - 2 \int t \cos t d t, t = \sqrt{4 - x} \end{aligned}$
$\begin{array}{l} = - 2 (\sqrt{4 - x} \sin \sqrt{4 - x} + \cos \sqrt{4 - x}) + C \\ - 20 = f (4) = - 2 + C \Rightarrow C = - 18 \end{array}$

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