Integral of (ln(x)e^ln(x))/x

\int\frac{\ln\left(x\right)e^{\ln\left(x\right)}}{x}dx

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e
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ln
log
lim
d/dx
d/dx
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sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
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sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Answer

$-x+x\ln\left(x\right)+C_0$

Step by step solution

Problem

$\int\frac{\ln\left(x\right)e^{\ln\left(x\right)}}{x}dx$
1

Simplifying the logarithm

$\int\frac{x\ln\left(x\right)}{x}dx$
2

Simplifying the fraction by $x$

$\int\ln\left(x\right)dx$
3

The integral of the natural logarithm is given by the following formula, $\displaystyle\int\ln(x)dx=x\ln(x)-x$

$x\ln\left(x\right)-x$
4

Factoring by $x$

$x\left(\ln\left(x\right)-1\right)$
5

Multiply $\left(\ln\left(x\right)+-1\right)$ by $x$

$x\ln\left(x\right)-x$
6

Add the constant of integration

$-x+x\ln\left(x\right)+C_0$

Answer

$-x+x\ln\left(x\right)+C_0$

Problem Analysis

Main topic:

Integral calculus

Time to solve it:

0.24 seconds

Views:

104