int {log x(log x + 2) , dx} = | vedclass

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

Question

(A) Let $I = \int {\log x(\log x + 2) \, dx}$.Substitute $\log x = t$,which implies $x = e^t$ and $dx = e^t \, dt$.Then,$I = \int t(t + 2) e^t \, dt =

Vedclass · Accepted Answer

$x{(\log x)^2} + c$

Vedclass · Answer

(A) Let $I = \int {\log x(\log x + 2) \, dx}$.Substitute $\log x = t$,which implies $x = e^t$ and $dx = e^t \, dt$.Then,$I = \int t(t + 2) e^t \, dt = \int (t^2 + 2t) e^t \, dt$.We know that $\int e^t [f(t) + f'(t)] \, dt = e^t f(t) + c$.Here,$f(t) = t^2$ and $f'(t) = 2t$.Therefore,$I = e^t \cdot t^2 + c$.Substituting back $t = \log x$ and $e^t = x$,we get $I = x{(\log x)^2} + c$.

$\int {\log x(\log x + 2) \, dx} = $

Similar Questions

$\int e^{-x}(x^3-2x^2+3x-4) dx=$

The value of $\int {{e^{2x}}(2\sin 3x + 3\cos 3x)\,dx} $ is

The value of $\int \frac{x e^{x} d x}{(1+x)^{2}}$ is equal to

Evaluate the integral: $\int \frac{x e^{2x}}{(1+2x)^2} dx = $ (where $C$ is a constant of integration.)

$\int \frac{(x-3) e^x}{(x-1)^3} d x=$ . . . . . . $+C$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module