int {left( {cos frac{x}{2} - sin frac{x}{2}} | vedclass

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Question

(A) We are given the integral: $\int {\left( {\cos \frac{x}{2} - \sin \frac{x}{2}} \right)^2} dx$ Expanding the square using the identity $(a-b)^2 = a

Vedclass · Accepted Answer

$x + \cos x + c$

Vedclass · Answer

(A) We are given the integral: $\int {\left( {\cos \frac{x}{2} - \sin \frac{x}{2}} ight)^2} dx$ Expanding the square using the identity $(a-b)^2 = a^2 + b^2 - 2ab$: $= \int {\left( {\cos^2 \frac{x}{2} + \sin^2 \frac{x}{2} - 2\sin \frac{x}{2}\cos \frac{x}{2}} ight)} dx$ Using the trigonometric identities $\cos^2 	heta + \sin^2 	heta = 1$ and $2\sin 	heta \cos 	heta = \sin(2	heta)$: $= \int {(1 - \sin x)} dx$ Integrating term by term: $= \int 1 dx - \int \sin x dx$ $= x - (-\cos x) + c$ $= x + \cos x + c$

$\int {\left( {\cos \frac{x}{2} - \sin \frac{x}{2}} \right)^2} dx = $

Similar Questions

Consider the following Assertion $(A)$ and Reason $(R)$:
Assertion $(A)$: $\int \sqrt{x-3} \left(\sin^{-1}(\log x) + \cos^{-1}(\log x)\right) dx = \frac{\pi}{3}(x-3)^{3/2} + c$
Reason $(R)$: $\sin^{-1}(f(x)) + \cos^{-1}(f(x)) = \frac{\pi}{2}$ for $|f(x)| \le 1$
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