int_2^3 frac{log x}{x} d x= | vedclass

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

Question

(C) Let $I = \int_2^3 \frac{\log x}{x} d x$. Substitute $u = \log x$,then $du = \frac{1}{x} dx$. When $x = 2$,$u = \log 2$. When $x = 3$,$u = \log 3$.

Vedclass · Accepted Answer

$\frac{1}{2} \log 6 \log \frac{3}{2}$

Vedclass · Answer

(C) Let $I = \int_2^3 \frac{\log x}{x} d x$. Substitute $u = \log x$,then $du = \frac{1}{x} dx$. When $x = 2$,$u = \log 2$. When $x = 3$,$u = \log 3$. Thus,$I = \int_{\log 2}^{\log 3} u du = \left[ \frac{u^2}{2} ight]_{\log 2}^{\log 3}$. $I = \frac{1}{2} \left( (\log 3)^2 - (\log 2)^2 ight)$. Using the identity $a^2 - b^2 = (a+b)(a-b)$,we get: $I = \frac{1}{2} (\log 3 + \log 2)(\log 3 - \log 2)$. Since $\log a + \log b = \log(ab)$ and $\log a - \log b = \log(\frac{a}{b})$,$I = \frac{1}{2} \log(3 	imes 2) \log(\frac{3}{2}) = \frac{1}{2} \log 6 \log \frac{3}{2}$.

$\int_2^3 \frac{\log x}{x} d x=$

Similar Questions

The approximate value of $\int_2^{10} x^2 dx$ by using the trapezoidal rule with $4$ equal intervals is:

If $f(x) = \frac{|\log x|}{x^2}$, then $\int_{1/e}^e f(x) dx =$

$\int_{-1}^{0} \frac{dx}{x^2 + 2x + 2} = $

If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral $\int_0^2 {{x^2}[x]\,dx} $ equals (in $/3$)

$\int_0^1 \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module