int_0^pi frac{x tan x}{sec x + tan x} ,dx = | vedclass

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

Question

(D) ધારો કે $I = \int_0^\pi \frac{x 	an x}{\sec x + 	an x} \,dx$. ગુણધર્મ $\int_0^a f(x) \,dx = \int_0^a f(a-x) \,dx$ નો ઉપયોગ કરતા: $I = \int_0^\pi

Vedclass · Accepted Answer

$\pi \left( \frac{\pi}{2} - 1 \right)$

Vedclass · Answer

(D) ધારો કે $I = \int_0^\pi \frac{x 	an x}{\sec x + 	an x} \,dx$. ગુણધર્મ $\int_0^a f(x) \,dx = \int_0^a f(a-x) \,dx$ નો ઉપયોગ કરતા: $I = \int_0^\pi \frac{(\pi - x) 	an(\pi - x)}{\sec(\pi - x) + 	an(\pi - x)} \,dx = \int_0^\pi \frac{(\pi - x)(-	an x)}{-\sec x - 	an x} \,dx = \int_0^\pi \frac{(\pi - x) 	an x}{\sec x + 	an x} \,dx$. $I$ માટેના બંને પદોનો સરવાળો કરતા: $2I = \int_0^\pi \frac{\pi 	an x}{\sec x + 	an x} \,dx = \pi \int_0^\pi \frac{\sin x}{1 + \sin x} \,dx$. $2I = \pi \int_0^\pi \frac{1 + \sin x - 1}{1 + \sin x} \,dx = \pi \left( \int_0^\pi 1 \,dx - \int_0^\pi \frac{1}{1 + \sin x} \,dx ight)$. અહીં $\int_0^\pi \frac{1}{1 + \sin x} \,dx = \int_0^\pi \frac{1 - \sin x}{\cos^2 x} \,dx = \int_0^\pi (\sec^2 x - \sec x 	an x) \,dx = [	an x - \sec x]_0^\pi = (0 - (-1)) - (0 - 1) = 2$. આમ,$2I = \pi (\pi - 2) = \pi^2 - 2\pi$. તેથી,$I = \frac{\pi^2}{2} - \pi = \pi \left( \frac{\pi}{2} - 1 ight)$.

$\int_0^\pi \frac{x \tan x}{\sec x + \tan x} \,dx = $

Similar Questions

ધારો કે $f(x) = \frac{x}{(1+x^n)^{1/n}}$,$x \in R - \{-1\}$,$n \in N$,$n > 2$. જો $f^n(x) = (f \circ f \circ f \dots n \text{ વખત})(x)$ હોય,તો $\lim_{n \to \infty} \int_0^1 x^{n-2} (f^n(x)) dx$ ની કિંમત $...............$ છે.

ધારો કે $I = \int_{0}^{100 \pi} \sqrt{1 - \cos 2x} \, dx$,તો

$\int_0^\pi \frac{x \tan x}{\sec x+\tan x} d x$ ની કિંમત શોધો.

$\int_0^\pi \sin^2 x \cos^3 x \, dx = $ . . . . . . .

જો સંકલન $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{323}}}\right) d x=\frac{\pi}{4}(\pi+a)-2$ નું મૂલ્ય હોય,તો $a$ નું મૂલ્ય શોધો.

Vedclass Test Series

Exam Paper Generator

Online Exam Module