Integrate the following functions- -tan x-sin x -cos x d x-

∫√(tan x)/sin x. cos xdx=∫√(tan x)/sin x cos x ×cos x/cos xdx [Multiply by 1 =cos x/cos x in denominator] =∫√(tan x)/tan x.cos^2 xdx =∫√(tan x)/tan x. sec^2 x d ...

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Byju's Answer

Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

Integrate the...

Question

Integrate the following functions.
$\int \frac{\sqrt{t a n x}}{s i n x . c o s x} d x .$

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Solution

$\int \frac{\sqrt{t a n x}}{s i n x . c o s x} d x = \int \frac{\sqrt{t a n x}}{s i n x c o s x \times \frac{c o s x}{c o s x}} d x$
[Multiply by $1 = \frac{c o s x}{c o s x}$ in denominator]
$= \int \frac{\sqrt{t a n x}}{t a n x . c o s^{2} x} d x = \int \frac{\sqrt{t a n x}}{t a n x} . s e c^{2} x d x$
Let $t a n x = t \Rightarrow s e c^{2} x = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{s e c^{2} x}$
$∴ \int \frac{\sqrt{t a n x}}{t a n x} . s e c^{2} x d x = \int \frac{\sqrt{t}}{t} . s e c^{2} x \frac{d t}{s e c^{2} x} = \int \frac{1}{\sqrt{t}} d t = \int t^{- \frac{1}{2}} d t = \frac{t^{- \frac{1}{2} + 1}}{(- \frac{1}{2} + 1)} + C = 2 \sqrt{t} + C = 2 \sqrt{t a n x} + C$

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Integrate the following functions. ∫√tanxsinx.cosxdx.

∫√tanxsinx.cosxdx=∫√tanxsinxcosx×cosxcosxdx [Multiply by 1=cosxcosx in denominator] =∫√tanxtanx.cos2xdx=∫√tanxtanx.sec2xdx Let tanx=t⇒sec2x=dtdx⇒dx=dtsec2x ∴∫√tanxtanx.sec2xdx=∫√tt.sec2xdtsec2x=∫1√tdt=∫t−12dt=t−12+1(−12+1)+C=2√t+C=2√tanx+C

Integrate the following functions.
$\int \frac{\sqrt{t a n x}}{s i n x . c o s x} d x .$