int operatorname{cosec}(x-a) cdot operatornam | vedclass

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

Question

(C) Let $I = \int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x \, dx$. We can write this as $I = \int \frac{1}{\sin (x-a) \sin x} \, dx$. Mu

Vedclass · Accepted Answer

$\frac{1}{\sin a} \log |\sin (x-a) \cdot \operatorname{cosec} x| + c$

Vedclass · Answer

(C) Let $I = \int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x \, dx$. We can write this as $I = \int \frac{1}{\sin (x-a) \sin x} \, dx$. Multiply and divide by $\sin a$: $I = \frac{1}{\sin a} \int \frac{\sin a}{\sin (x-a) \sin x} \, dx$. Express $a$ as $x - (x-a)$: $I = \frac{1}{\sin a} \int \frac{\sin (x - (x-a))}{\sin (x-a) \sin x} \, dx$. Using the formula $\sin(A-B) = \sin A \cos B - \cos A \sin B$: $I = \frac{1}{\sin a} \int \frac{\sin x \cos (x-a) - \cos x \sin (x-a)}{\sin (x-a) \sin x} \, dx$. $I = \frac{1}{\sin a} \int \left( \frac{\sin x \cos (x-a)}{\sin (x-a) \sin x} - \frac{\cos x \sin (x-a)}{\sin (x-a) \sin x} \right) \, dx$. $I = \frac{1}{\sin a} \int (\cot (x-a) - \cot x) \, dx$. Integrating,we get: $I = \frac{1}{\sin a} [\log |\sin (x-a)| - \log |\sin x|] + c$. Using the property $\log m - \log n = \log(\frac{m}{n})$: $I = \frac{1}{\sin a} \log \left| \frac{\sin (x-a)}{\sin x} \right| + c$. Since $\frac{1}{\sin x} = \operatorname{cosec} x$,this is equivalent to: $I = \frac{1}{\sin a} \log |\sin (x-a) \cdot \operatorname{cosec} x| + c$.

$\int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x \, dx = $

Similar Questions

$\int \frac{1}{\sqrt{8+2x-x^2}} dx =$

$\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} \, dx$

Evaluate the integral: $\int \left(\frac{8^{1+x}+4^{1+x}}{2^{2x}}\right) dx$

Integrate the function: $\frac{1}{\sqrt{7-6x-x^{2}}}$

The value of $\int \frac{2 x^3-1}{x^4+x} \,d x$ is equal to (where $C$ is a constant of integration.)

Vedclass Test Series

Exam Paper Generator

Online Exam Module