If f^{prime}(x)=tan ^{-1}(sec x+tan x),-frac{ | vedclass

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

Question

(A) Given $f^{\prime}(x) = 	an ^{-1}(\sec x + 	an x)$.Simplifying the expression inside the inverse tangent:$f^{\prime}(x) = 	an ^{-1}\left(\frac{1

Vedclass · Accepted Answer

$\frac{\pi+1}{4}$

Vedclass · Answer

(A) Given $f^{\prime}(x) = 	an ^{-1}(\sec x + 	an x)$.Simplifying the expression inside the inverse tangent:$f^{\prime}(x) = 	an ^{-1}\left(\frac{1+\sin x}{\cos x}ight)$Using half-angle identities $\sin x = 2\sin\frac{x}{2}\cos\frac{x}{2}$,$\cos x = \cos^2\frac{x}{2} - \sin^2\frac{x}{2}$,and $1 = \cos^2\frac{x}{2} + \sin^2\frac{x}{2}$:$f^{\prime}(x) = 	an ^{-1}\left[\frac{(\cos \frac{x}{2} + \sin \frac{x}{2})^2}{(\cos \frac{x}{2} - \sin \frac{x}{2})(\cos \frac{x}{2} + \sin \frac{x}{2})}ight]$$f^{\prime}(x) = 	an ^{-1}\left(\frac{\cos \frac{x}{2} + \sin \frac{x}{2}}{\cos \frac{x}{2} - \sin \frac{x}{2}}ight)$Dividing numerator and denominator by $\cos \frac{x}{2}$:$f^{\prime}(x) = 	an ^{-1}\left(\frac{1 + 	an \frac{x}{2}}{1 - 	an \frac{x}{2}}ight) = 	an ^{-1}\left[	an \left(\frac{\pi}{4} + \frac{x}{2}ight)ight]$Thus,$f^{\prime}(x) = \frac{\pi}{4} + \frac{x}{2}$.Integrating with respect to $x$:$f(x) = \int \left(\frac{\pi}{4} + \frac{x}{2}ight) dx = \frac{\pi x}{4} + \frac{x^2}{4} + C$.Given $f(0) = 0$,we get $C = 0$.So,$f(x) = \frac{\pi x + x^2}{4}$.Therefore,$f(1) = \frac{\pi(1) + (1)^2}{4} = \frac{\pi+1}{4}$.

If $f^{\prime}(x)=\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{2} < x < \frac{\pi}{2}$ and $f(0)=0$,then $f(1)$ is

Similar Questions

If $f^{\prime}(x)=\tan^{-1}(\sec x+\tan x)$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$ and $f(0)=0$,then $f(1)$ is equal to:

The derivative of $f(x) = x|x|$ is

If $f(x) = \frac{e^{2x} - e^{-2x}}{e^{3x} + e^{-3x}}$,then $f^{\prime}(0) = $

If $G(x) = -\sqrt{25-x^{2}}$,then $\lim _{x \rightarrow 1} \frac{G(x)-G(1)}{x-1}$ is

If $f(x)$ is an even function,then $f^{\prime}(x)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module