Integrate the following functions- -sin 2 xcos 2 x d x

∫√(sin 2x) cos 2x dx Let sin 2x=t ⇒ 2 cos 2x =dt/dx⇒ dx =dt/2 cos 2x ∴∫√(sin 2x) cos 2x dx =∫√(t) cos 2x dt/2 cos 2x=1/2∫√(t)dt =1/2t^1/2+1/(1/2+1)+C =1/3t^3/2 ...

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Standard XII

Mathematics

Functions

Integrate the...

Question

Integrate the following functions.
$\int \sqrt{s i n 2 x} c o s 2 x d x .$

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Solution

$\int \sqrt{s i n 2 x} c o s 2 x d x$
Let $s i n 2 x = t \Rightarrow 2 c o s 2 x = \frac{d t}{d x} \Rightarrow d x = \frac{d t}{2 c o s 2 x}$
$∴ \int \sqrt{s i n 2 x} c o s 2 x d x = \int \sqrt{t} c o s 2 x \frac{d t}{2 c o s 2 x} = \frac{1}{2} \int \sqrt{t} d t = \frac{1}{2} \frac{t^{\frac{1}{2} + 1}}{(\frac{1}{2} + 1)} + C = \frac{1}{3} t^{\frac{3}{2}} + C = \frac{1}{3} (s i n 2 x)^{\frac{3}{2}} + C$

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Integrate the following functions. ∫√sin2xcos2xdx.

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