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

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

Question

(D) Let $I = \int \frac{\log \sqrt{x}}{3 x} \,d x$.We know that $\log \sqrt{x} = \log (x^{1/2}) = \frac{1}{2} \log x$.Substituting this into the integ

Vedclass · Accepted Answer

$\frac{1}{12}(\log x)^2+c$,(where $c$ is a constant of integration)

Vedclass · Answer

(D) Let $I = \int \frac{\log \sqrt{x}}{3 x} \,d x$.We know that $\log \sqrt{x} = \log (x^{1/2}) = \frac{1}{2} \log x$.Substituting this into the integral:$I = \int \frac{\frac{1}{2} \log x}{3 x} \,d x = \frac{1}{6} \int \frac{\log x}{x} \,d x$.Let $u = \log x$,then $du = \frac{1}{x} \,d x$.The integral becomes $I = \frac{1}{6} \int u \,du = \frac{1}{6} \cdot \frac{u^2}{2} + c = \frac{u^2}{12} + c$.Substituting $u = \log x$ back,we get $I = \frac{1}{12} (\log x)^2 + c$.

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

Similar Questions

$\int \frac{d x}{(x+1) \sqrt{4 x+3}}$ is equal to

$\int \frac{d x}{\sqrt{x}(x+9)}$ is equal to

If $\int \log \left(6 \sin ^2 x+17 \sin x+12\right)^{\cos x} d x=f(x)+c$ then,$f\left(\frac{\pi}{2}\right)=$

If $I=\int \frac{dx}{x^2(x^4+1)^{3/4}}$,then $I$ is

Evaluate $\int \sin (\sqrt{k}) \, dk$ on $(0, \infty)$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module