Let 2^{1-a} + 2^{1+a}, f(a), 3^a + 3^{-a} be | 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 $2f(a) = (2^{1-a} + 2^{1+a}) + (3^a + 3^{-a})$.Using the $AM$-$

Vedclass · Accepted Answer

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

Vedclass · Answer

(B) Given that $2^{1-a} + 2^{1+a}$, $f(a)$, and $3^a + 3^{-a}$ are in $A$.$P$.,we have $2f(a) = (2^{1-a} + 2^{1+a}) + (3^a + 3^{-a})$.Using the $AM$-$GM$ inequality, $2^{1-a} + 2^{1+a} = 2(2^{-a} + 2^a) \geq 2(2) = 4$ and $3^a + 3^{-a} \geq 2$. Both reach their minimum at $a=0$.Thus, the minimum value $\alpha$ of $f(a)$ is $\alpha = \frac{1}{2}(4 + 2) = 3$.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 $t = e^{2x}$, then $dt = 2e^{2x} dx$, so $e^{2x} dx = \frac{dt}{2}$.When $x = \log_e(2)$, $t = e^{2\log_e(2)} = 4$. When $x = \log_e(3)$, $t = e^{2\log_e(3)} = 9$.$I = \frac{1}{2} \int_{4}^{9} \frac{dt}{t^2 - 1} = \frac{1}{2} \cdot \frac{1}{2} [\log_e|\frac{t-1}{t+1}|]_{4}^{9} = \frac{1}{4} [\log_e(\frac{8}{10}) - \log_e(\frac{3}{5})] = \frac{1}{4} \log_e(\frac{8/10}{3/5}) = \frac{1}{4} \log_e(\frac{4}{3})$.

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

If $\frac{d}{{dx}}G(x) = \frac{{{e^{\tan x}}}}{x}$ for $x \in (0, \pi/2)$,then $\int_{1/4}^{1/2} \frac{2}{x} e^{\tan(\pi x^2)} dx$ is equal to

The value of $\int_0^{\pi /2} \frac{\sin x}{1 + \cos^2 x} \, dx$ is

$\int_{1}^{e} \frac{dx}{x(1+\log x)^{2}} =$

Evaluate the definite integral $\int_{0}^{1} \frac{x}{x^{2}+1} d x$.

Evaluate $\int_{\sin \theta}^{\cos \theta} f(x \tan \theta) \, dx$,where $\theta \neq \frac{n \pi}{2}, n \in I$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module