First slide

Introduction to integration

Question

$Evaluate \int \frac{1}{\sqrt{3} \sin x + \cos x} d x$

Easy

A

$\frac{1}{2} \log |\tan (\frac{x}{2} + \frac{π}{6})| + c$
B

$\frac{1}{2} \log |\tan (\frac{x}{2} + \frac{π}{12})| + c$
C

$\frac{1}{2} \log |\tan (\frac{x}{2})| + c$
D

None of these

Solution

$Let \sqrt{3} = r \sin θ and 1 = r \cos θ Then r = \sqrt{(\sqrt{3})^{2} + 1^{2}} = 2 and \tan θ = \frac{\sqrt{3}}{1} or θ = \frac{π}{3}$
$\begin{array}{l} ∴ \int \frac{1}{\sqrt{3} \sin x + \cos x} d x \\ = \int \frac{1}{r \sin θ \sin x + r \cos θ \cos x} d x \\ = \frac{1}{r} \int \frac{1}{\cos (x - θ)} d x = \frac{1}{r} \int \sec (x - θ) d x \end{array}$
$\begin{array}{ll} = \frac{1}{r} \log |\tan (\frac{π}{4} + \frac{x}{2} - \frac{θ}{2})| + c \\ = \frac{1}{2} \log |\tan (\frac{π}{4} + \frac{x}{2} - \frac{π}{6})| + c \\ = \frac{1}{2} \log |\tan (\frac{x}{2} + \frac{π}{12})| + c \end{array}$

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