First slide

Introduction to integration

Question

The integral of $f (x) = 3^{x} \cos x$ is given by

Easy

A

$3^{x} (\sin x + \cos x \log 3) + C$
B

$3^{x} (\log 3 \sin x + \cos x) + C$
C

$\frac{3^{x} (\sin x + \cos \log 3)}{(1 + (\log 3)^{2})}$
D

$\frac{3^{x} (\log 3 \sin x + \cos x)}{(1 + (\log 3)^{2})} + C$

Solution

$\begin{array}{l} I = \int (3^{x} \cos x d x \\ = 3^{x} \sin x - \log 3 \int 3^{x} \sin x d x \\ = 3^{x} \sin x - \log 3 [- 3^{x} \cos x + (\log 3) \int 3^{x} \end{array} \cos x d x]$
$\begin{matrix} = 3^{x} \sin x + (\log 3), 3^{x} \cos x - (\log 3)^{2} \\ \Rightarrow (1 + (\log 3)^{2}) I = 3^{x} (\sin x + (\log 3) \cos x) \\ I = \frac{3^{x} (\sin x + (\log 3) \cos x)}{(1 + (\log 3)^{2})} \end{matrix}$

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