If int frac{dx}{cos^3 x sqrt{2 sin 2x}} = (ta | vedclass

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

Question

(A) We have the integral $I = \int \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}}$.Using $\sin 2x = 2 \sin x \cos x$,the integral becomes:$I = \int \frac{dx}{\c

Vedclass · Accepted Answer

$\frac{16}{5}$

Vedclass · Answer

(A) We have the integral $I = \int \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}}$.Using $\sin 2x = 2 \sin x \cos x$,the integral becomes:$I = \int \frac{dx}{\cos^3 x \sqrt{4 \sin x \cos x}} = \int \frac{dx}{2 \cos^3 x \sqrt{\sin x \cos x}} = \int \frac{dx}{2 \cos^4 x \sqrt{	an x}}$.Since $\frac{1}{\cos^4 x} = \sec^4 x = (1 + 	an^2 x) \sec^2 x$,we have:$I = \frac{1}{2} \int \frac{(1 + 	an^2 x) \sec^2 x}{\sqrt{	an x}} dx$.Let $u = 	an x$,then $du = \sec^2 x dx$:$I = \frac{1}{2} \int \frac{1 + u^2}{\sqrt{u}} du = \frac{1}{2} \int (u^{-1/2} + u^{3/2}) du$.Integrating term by term:$I = \frac{1}{2} \left( \frac{u^{1/2}}{1/2} + \frac{u^{5/2}}{5/2} ight) + k = u^{1/2} + \frac{1}{5} u^{5/2} + k$.Substituting $u = 	an x$ back:$I = (	an x)^{1/2} + \frac{1}{5} (	an x)^{5/2} + k$.Comparing this with the given form $(	an x)^A + C(	an x)^B + k$,we get $A = 1/2$,$B = 5/2$,and $C = 1/5$.Therefore,$A + B + C = \frac{1}{2} + \frac{5}{2} + \frac{1}{5} = 3 + \frac{1}{5} = \frac{16}{5}$.

If $\int \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} = (\tan x)^A + C(\tan x)^B + k,$ where $k$ is a constant of integration,then $A + B + C$ equals

Similar Questions

In the reaction $A + 2B \rightleftharpoons 2C + D$,the initial concentration of $B$ was $1.5$ times that of $[A]$. At equilibrium,the concentrations of $A$ and $B$ became equal. The equilibrium constant $K_c$ for the reaction is:

If $\frac{x^3}{(2x-1)(x+2)(x-3)} = A + \frac{B}{2x-1} + \frac{C}{x+2} + \frac{D}{x-3}$,then $A$ is equal to

When electric current is passed through acidified water for $1930 \ s$,$1120 \ mL$ of $H_2$ gas is collected (at $STP$) at the cathode. What is the current passed in amperes?

In Young's double-slit experiment,the intensity at a point where the path difference is $\lambda / 6$ is $I'$. If $I_0$ denotes the maximum intensity,then $I'/I_0$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module