∫((cos^(1.5) x−sin^(1.5) x)/( (√(sinx cosx)))) dx = ∫((cos^(3/2) x)/(sin^(1/2) x cos^(1/2) x))dx−∫((sin^(3/2) x)/(sin^(1/2) x cos^(1/2) x)) dx = ∫ ((cosx)/(sin^(1/2) x)) dx − ∫ ((sinx)/(cos^(1/2) x)) dx = ∫ (dt/t^(1/2) ) − ∫ (((−dz))/z^(1/2) ) where cosx = z and sinx = t = 2 (√(sinx)) + 2 (√(cosx)) + C


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Question Number 190452 by vishal1234 last updated on 03/Apr/23

$\int \frac{{c o s}^{1.5} x - {s i n}^{1.5} x}{\sqrt{s i n x c o s x}} d x$ $= \int \frac{{c o s}^{\frac{3}{2}} x}{{s i n}^{1 / 2} x {c o s}^{1 / 2} x} d x - \int \frac{{s i n}^{\frac{3}{2}} x}{{s i n}^{1 / 2} x {c o s}^{1 / 2} x} d x$ $= \int \frac{c o s x}{{s i n}^{1 / 2} x} d x - \int \frac{s i n x}{{c o s}^{1 / 2} x} d x$ $= \int \frac{d t}{t^{1 / 2}} - \int \frac{(- d z)}{z^{1 / 2}}$ $w h e r e c o s x = z a n d s i n x = t$ $= 2 \sqrt{s i n x} + 2 \sqrt{c o s x} + C$
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