int frac{1+sqrt{3} cot x}{1-sqrt{3} cot x} d | vedclass

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

Question

(A) $\int \frac{1+\sqrt{3} \cot x}{1-\sqrt{3} \cot x} d x = \int \frac{	an x+\sqrt{3}}{	an x-\sqrt{3}} d x = \int \frac{\sin x+\sqrt{3} \cos x}{\sin

Vedclass · Accepted Answer

$-\frac{x}{2}+\frac{\sqrt{3}}{2} \log \left|\sin \left(x-\frac{\pi}{3}\right)\right|+c$

Vedclass · Answer

(A) $\int \frac{1+\sqrt{3} \cot x}{1-\sqrt{3} \cot x} d x = \int \frac{	an x+\sqrt{3}}{	an x-\sqrt{3}} d x = \int \frac{\sin x+\sqrt{3} \cos x}{\sin x-\sqrt{3} \cos x} d x$Let $\sin x+\sqrt{3} \cos x = K_1(\cos x+\sqrt{3} \sin x) + K_2(\sin x-\sqrt{3} \cos x)$.Comparing coefficients of $\sin x$ and $\cos x$:$\sqrt{3} K_1 + K_2 = 1$ $(i)$$K_1 - \sqrt{3} K_2 = \sqrt{3}$ (ii)Solving these,we get $K_1 = \frac{\sqrt{3}}{2}$ and $K_2 = -\frac{1}{2}$.Thus,the integral becomes $\int \frac{K_1(\cos x+\sqrt{3} \sin x) + K_2(\sin x-\sqrt{3} \cos x)}{\sin x-\sqrt{3} \cos x} d x$$= K_1 \int \frac{d(\sin x-\sqrt{3} \cos x)}{\sin x-\sqrt{3} \cos x} + K_2 \int 1 d x$$= \frac{\sqrt{3}}{2} \ln |\sin x-\sqrt{3} \cos x| - \frac{1}{2} x + C$Since $\sin x-\sqrt{3} \cos x = 2(\frac{1}{2} \sin x - \frac{\sqrt{3}}{2} \cos x) = 2 \sin(x-\frac{\pi}{3})$,The expression becomes $-\frac{x}{2} + \frac{\sqrt{3}}{2} \ln |2 \sin(x-\frac{\pi}{3})| + C = -\frac{x}{2} + \frac{\sqrt{3}}{2} \ln |\sin(x-\frac{\pi}{3})| + C'$.

$\int \frac{1+\sqrt{3} \cot x}{1-\sqrt{3} \cot x} d x=$

Similar Questions

For $n \geq 2$,let $I_n = \int_0^{\pi / 4} \tan^n x \, dx$ and $F_n = I_n + I_{n-2}$. Then,$F_n - F_{n+1} =$

For any integer $n \geq 2$,let $I_n = \int \tan^n x \, dx$. If $I_n = \frac{1}{a} \tan^{n-1} x - b I_{n-2}$ for $n \geq 2$,then the ordered pair $(a, b)$ is equal to

$\int \frac{(1-4 \sin^2 x) \cos x}{\cos (3x+2)} dx =$

$\int \frac{\cos^3 x + \cos^5 x}{\sin^2 x + \sin^4 x} dx =$

Observe the following statements :
$A: \int \left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x = e^{\frac{x^2+1}{x}} + c$
$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$
Then which of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module