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Trigonometric Ratios for Sum of Two Angles

Integrate the...

Question

Integrate the following functions.
$\int \frac{1}{(1 + c o t x)} d x .$

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Solution

Let $I = \int \frac{1}{(1 + c o t x)} d x = \int \frac{1}{1 + \frac{c o s x}{s i n x}} d x$
$= \int \frac{1}{\frac{s i n x + c o s x}{s i n x}} d x = \int \frac{s i n x}{(s i n x + c o s x)} d x = \frac{1}{2} \int \frac{2 (s i n x)}{s i n x + c o s x} d x = \frac{1}{2} \int \frac{s i n x + s i n x + c o s x - c o s x}{s i n x + c o s x} d x$
[add and subtract cos x in numerator]
$= \frac{1}{2} \int \frac{(s i n x + c o s x) + (s i n x - c o s x)}{(s i n x + c o s x)} d x = \frac{1}{2} [\int \frac{(s i n x + c o s x)}{(s i n x + c o s x)} d x + \int \frac{(s i n x - c o s x)}{s i n x + c o s x} d x] = \frac{1}{2} [\int 1 d x + \int \frac{(s i n x - c o s x)}{s i n x + c o s x} d x]$
Let $s i n x + c o s x = t \Rightarrow c o s x - s i n x = \frac{d t}{d x}$
$\Rightarrow d x = \frac{d t}{- [s i n x - c o s x]} ∴ I = \frac{1}{2} [\int 1 d x + \int \frac{(s i n x - c o s x)}{t} \frac{d t}{- [s i n x - c o s x]}] = \frac{1}{2} [\int 1 d x - \int \frac{1}{t} d t] = \frac{1}{2} [x - l o g | t |] + C = \frac{1}{2} [x - l o g | s i n x + c o s x |] + C$

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Trigonometric Ratios for Sum of Two Angles

Standard XII Mathematics

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