int (log x)^m x^n , dx = | vedclass

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

Question

(C) Let $I = \int (\log x)^m x^n \, dx$. Substitute $\log x = t$,which implies $x = e^t$. Then,differentiating both sides with respect to $t$,we get $

Vedclass · Accepted Answer

$\int t^m e^{(n+1)t} \, dt, x = e^t$

Vedclass · Answer

(C) Let $I = \int (\log x)^m x^n \, dx$. Substitute $\log x = t$,which implies $x = e^t$. Then,differentiating both sides with respect to $t$,we get $dx = e^t \, dt$. Substituting these into the integral: $I = \int t^m (e^t)^n \cdot e^t \, dt$ $I = \int t^m e^{nt} \cdot e^t \, dt$ $I = \int t^m e^{(n+1)t} \, dt$. Thus,the correct substitution is $x = e^t$ and the resulting integral is $\int t^m e^{(n+1)t} \, dt$.

$\int (\log x)^m x^n \, dx =$

Similar Questions

If $\int {\frac{{({x^2} - 1)\,dx}}{{({x^4} + 3{x^2} + 1)\,{{\tan }^{ - 1}}\left( {\frac{{{x^2} + 1}}{x}} \right)}}} = \ln | f(x) | + C$,then $f(x)$ is:

$\int {{e^{{{\cos }^2}x}}\sin 2x\,dx = }$

$\int \frac{(3 x-2) \tan \left(\sqrt{9 x^2-12 x+1}\right)}{\sqrt{9 x^2-12 x+1}} d x=$

If $\int(2x+4)\sqrt{x-1}dx = a(x-1)^{5/2} + b(x-1)^{3/2} + c$ where $c$ is a constant of integration,then the value of $(2a+b)$ is

$\int \frac{\sqrt{x}}{x+1} \,dx=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module