Find the integrals of the functions-cos x-sin x-1-sin 2 x d x

∫cos x sin x/1+sin 2xdx. Let I =∫cos x sin x/1+sin 2xdx =∫cos x sin x/sin^2 x +cos^2 x+2 sin x cos xdx [∵ sin^2 x+cos^2 x=1. sin 2x=2 sin x cos x] =∫cos x sin ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals - 1

Find the inte...

Question

Find the integrals of the functions.
$\int \frac{c o s x - s i n x}{1 + s i n 2 x} d x .$

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Solution

$\int \frac{c o s x - s i n x}{1 + s i n 2 x} d x .$
Let $I = \int \frac{c o s x - s i n x}{1 + s i n 2 x} d x = \int \frac{c o s x - s i n x}{s i n^{2} x + c o s^{2} x + 2 s i n x c o s x} d x$
$[∵ s i n^{2} x + c o s^{2} x = 1. s i n 2 x = 2 s i n x c o s x] = \int \frac{c o s x - s i n x}{(s i n x + c o s x)^{2}} d x L e t c o s x + s i n x = t \Rightarrow - s i n x + c o s x = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{(c o s x - s i n x)} ∴ I = \int \frac{c o s x - s i n x}{t^{2}} . \frac{d t}{(c o s x - s i n x)} = \int \frac{1}{t^{2}} = \int t^{- 2} d t = \frac{t^{- 2 + 1}}{- 2 + 1} + C = \frac{- 1}{c o s x + s i n x} + C$

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Find the integrals of the functions. ∫cosx−sinx1+sin2xdx.

Find the integrals of the functions.
$\int \frac{c o s x - s i n x}{1 + s i n 2 x} d x .$