If int e^{alpha x}left(frac{1-beta sin x}{1-c | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given, $\int e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) d x=-e^x \cot \frac{x}{2}+c$. By differentiating both sides with respect to

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) Given, $\int e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) d x=-e^x \cot \frac{x}{2}+c$. By differentiating both sides with respect to $x$, we get: $e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) = \frac{d}{dx} \left(-e^x \cot \frac{x}{2}\right) + 0$. Using the product rule, $\frac{d}{dx}(uv) = u'v + uv'$: $e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) = -\left[e^x \cot \frac{x}{2} + e^x \left(-\frac{1}{2} \csc^2 \frac{x}{2}\right)\right]$. $e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) = -e^x \left[\frac{\cos(x/2)}{\sin(x/2)} - \frac{1}{2 \sin^2(x/2)}\right]$. $e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) = -e^x \left[\frac{2 \sin(x/2) \cos(x/2) - 1}{2 \sin^2(x/2)}\right]$. Using $2 \sin(x/2) \cos(x/2) = \sin x$ and $2 \sin^2(x/2) = 1 - \cos x$: $e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) = -e^x \left[\frac{\sin x - 1}{1 - \cos x}\right] = e^x \left(\frac{1 - \sin x}{1 - \cos x}\right)$. Comparing both sides, we get $\alpha = 1$ and $\beta = 1$. Therefore, $\frac{\alpha^2 + \beta^2}{2 \alpha \beta} = \frac{1^2 + 1^2}{2(1)(1)} = \frac{2}{2} = 1$.

If $\int e^{\alpha x}\left(\frac{1-\beta \sin x}{1-\cos x}\right) d x=-e^x \cot \frac{x}{2}+c$, then $\frac{\alpha^2+\beta^2}{2 \alpha \beta}=$

Similar Questions

Integrate the function: $e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)$

$\int_{\alpha+1}^{\alpha} \frac{e^x(\alpha-x)}{(x-\alpha+1)^2} dx =$

$\int \frac{x-3}{(x-1)^3} e^x \, dx =$

$\int {{e^x} \left( {\frac{1}{x} - \frac{1}{{{x^2}}}} \right)} \,dx = $

$\int\limits_1^2 {{e^{2x}}} \left( {\frac{1}{x} - \frac{1}{{2{x^2}}}} \right)\,dx$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module