int frac{log sqrt{x}}{3 x} dx is equal to | vedclass

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

Question

(A) Let $I = \int \frac{\log \sqrt{x}}{3 x} dx$. Using the property $\log \sqrt{x} = \log(x^{1/2}) = \frac{1}{2} \log x$,the integral becomes: $I = \i

Vedclass · Accepted Answer

$\frac{1}{3}(\log \sqrt{x})^2+C$

Vedclass · Answer

(A) Let $I = \int \frac{\log \sqrt{x}}{3 x} dx$. Using the property $\log \sqrt{x} = \log(x^{1/2}) = \frac{1}{2} \log x$,the integral becomes: $I = \int \frac{\frac{1}{2} \log x}{3 x} dx = \frac{1}{6} \int \frac{\log x}{x} dx$. Let $u = \log x$,then $du = \frac{1}{x} dx$. Substituting these into the integral: $I = \frac{1}{6} \int u du = \frac{1}{6} \cdot \frac{u^2}{2} + C = \frac{u^2}{12} + C$. Substituting $u = \log x$ back: $I = \frac{(\log x)^2}{12} + C$. Alternatively,using the substitution $t = \log \sqrt{x} = \frac{1}{2} \log x$,then $dt = \frac{1}{2x} dx$,which implies $\frac{1}{x} dx = 2 dt$. $I = \int \frac{t}{3} (2 dt) = \frac{2}{3} \int t dt = \frac{2}{3} \cdot \frac{t^2}{2} + C = \frac{t^2}{3} + C = \frac{1}{3} (\log \sqrt{x})^2 + C$. Thus,the correct option is $A$.

$\int \frac{\log \sqrt{x}}{3 x} dx$ is equal to

Similar Questions

In the isoelectronic series of metal carbonyls,the $CO$ bond strength is expected to increase in the order:

Describe the bulk preparation of dihydrogen by the electrolytic method. What is the role of an electrolyte in this process?

Which one of the following population interactions is widely used in medical science for the production of antibiotics?

The pulley and spring are massless,and friction is absent everywhere. The $5\,kg$ block is released from rest. The speed of the $5\,kg$ block when the $2\,kg$ block leaves contact with the ground is (take the force constant of the spring $K = 40\,N/m$ and $g = 10\,m/s^2$):

The numbers $a_n = 6^n - 5n$ for $n = 1, 2, 3, \ldots$ when divided by $25$ leave the remainder:

Vedclass Test Series

Exam Paper Generator

Online Exam Module