If int {{e^{sec x}}left( {sec x + tan x f(x) | vedclass

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

Question

(D) Given the integral equation: $\int e^{\sec x} (\sec x + 	an x f(x) + \sec x 	an x + \sec^2 x) dx = e^{\sec x} f(x) + C$. Differentiating both si

Vedclass · Accepted Answer

$\sec x + 	an x + \frac{1}{2}$

Vedclass · Answer

(D) Given the integral equation: $\int e^{\sec x} (\sec x + 	an x f(x) + \sec x 	an x + \sec^2 x) dx = e^{\sec x} f(x) + C$. Differentiating both sides with respect to $x$ using the Fundamental Theorem of Calculus: $e^{\sec x} (\sec x + 	an x f(x) + \sec x 	an x + \sec^2 x) = \frac{d}{dx} (e^{\sec x} f(x))$. Applying the product rule on the right side: $e^{\sec x} (\sec x + 	an x f(x) + \sec x 	an x + \sec^2 x) = e^{\sec x} \cdot \sec x 	an x \cdot f(x) + e^{\sec x} f'(x)$. Dividing both sides by $e^{\sec x}$: $\sec x + 	an x f(x) + \sec x 	an x + \sec^2 x = \sec x 	an x f(x) + f'(x)$. Rearranging to solve for $f'(x)$: $f'(x) = \sec x + \sec x 	an x + \sec^2 x + 	an x f(x) - \sec x 	an x f(x)$. For the equation to hold,we set the terms involving $f(x)$ to cancel out or match the structure. If we assume $f(x) = \sec x + 	an x$,then $f'(x) = \sec x 	an x + \sec^2 x$. Substituting this into the equation: $\sec x + 	an x (\sec x + 	an x) + \sec x 	an x + \sec^2 x = \sec x 	an x (\sec x + 	an x) + (\sec x 	an x + \sec^2 x)$. This simplifies to an identity. Thus,$f(x) = \sec x + 	an x + c$. Comparing with the options,$f(x) = \sec x + 	an x + \frac{1}{2}$ is a valid choice.

If $\int {{e^{\sec x}}\left( {\sec x + \tan x f(x) + (\sec x \tan x + \sec^2 x)} \right)dx = {e^{\sec x}}f(x) + C}$,then a possible choice of $f(x)$ is

Similar Questions

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

If $\int e^x(x^3+x^2-x+4) dx = e^x f(x) + c$,then $f(1) =$

If $\int e^x(\sin^2 2x - 8 \cos 4x) dx = e^x f(x) + c$,then $f(\frac{\pi}{4}) = $

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

$\int e^x \left( \frac{2 + \sin 2x}{1 + \cos 2x} \right) dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module