Let (2^{1-a} + 2^{1+a}), f(a), (3^a + 3^{-a}) | vedclass

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

Question

(B) Given that $(2^{1-a} + 2^{1+a})$, $f(a)$, and $(3^a + 3^{-a})$ are in $A$.$P$.,we have $2f(a) = (2^{1-a} + 2^{1+a}) + (3^a + 3^{-a})$.Since $2^{1-

Vedclass · Accepted Answer

$\frac{1}{4}\log_e(\frac{4}{3})$

Vedclass · Answer

(B) Given that $(2^{1-a} + 2^{1+a})$, $f(a)$, and $(3^a + 3^{-a})$ are in $A$.$P$.,we have $2f(a) = (2^{1-a} + 2^{1+a}) + (3^a + 3^{-a})$.Since $2^{1-a} + 2^{1+a} = 2(2^{-a} + 2^a)$, and by the $AM$-$GM$ inequality, $2^a + 2^{-a} \geq 2$, the minimum value of $2^a + 2^{-a}$ is $2$.Similarly, the minimum value of $3^a + 3^{-a}$ is $2$.Thus, $2f(a) \geq 2(2) + 2 = 6$, which implies $f(a) \geq 3$. Therefore, $\alpha = 3$.The integral becomes $I = \int_{\log_e 2}^{\log_e 3} \frac{dx}{e^{2x} - e^{-2x}} = \int_{\log_e 2}^{\log_e 3} \frac{e^{2x} dx}{e^{4x} - 1}$.Let $u = e^{2x}$, then $du = 2e^{2x} dx$, so $e^{2x} dx = \frac{du}{2}$.When $x = \log_e 2$, $u = e^{2\log_e 2} = 4$. When $x = \log_e 3$, $u = e^{2\log_e 3} = 9$.$I = \frac{1}{2} \int_{4}^{9} \frac{du}{u^2 - 1} = \frac{1}{2} \cdot \frac{1}{2} \log_e |\frac{u-1}{u+1}| \Big|_4^9 = \frac{1}{4} (\log_e \frac{8}{10} - \log_e \frac{3}{5}) = \frac{1}{4} \log_e (\frac{4}{5} \cdot \frac{5}{3}) = \frac{1}{4} \log_e (\frac{4}{3})$.

Let $(2^{1-a} + 2^{1+a})$, $f(a)$, $(3^a + 3^{-a})$ be in $A$.$P$. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\int_{\log_e(\alpha-1)}^{\log_e(\alpha)} \frac{dx}{(e^{2x} - e^{-2x})}$ is:

Similar Questions

$\int_{0}^{2} x e^{x} dx =$

$\int_0^{\pi / 4} [\sqrt{\tan x} + \sqrt{\cot x}] \, dx$ is equal to

Let $a_{n} = \int_{-1}^{n} \left(1 + \frac{x}{2} + \frac{x^{2}}{3} + \ldots + \frac{x^{n-1}}{n}\right) dx$ for $n \in N$. Then the sum of all the elements of the set $\{n \in N : a_{n} \in (2, 30)\}$ is $...........$

Let $f(x) = \max \{x+|x|, x-[x]\}$, where $[x]$ denotes the greatest integer $\leq x$. Then, the value of $\int_{-3}^{3} f(x) dx$ is

Evaluate the definite integral $\int_{0}^{\frac{\pi}{2}} \cos ^{2} x \,d x$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module