Question

find the integration of 1 divided by x into 1+logx dx​

Answer 1

$\bf\large\underline\blue{Question:-}$

• Find the integration of $\int \frac{1 + log(x) }{x}\: dx$

$\bf\large\underline\blue{Solution:-}$

$\int\frac{1 + log(x) }{x}\: dx$

First of all, we will separate the fraction into two fractions,

$\int\frac{1}{x} + \frac{ log(x) }{x}\: dx$

Then, using the property of integrals,

$\int f(x) \pm g(x) \: dx = \int f(x) \: dx \pm \int g(x) \: dx$

We get,

$\int \frac{1}{x} \: dx + \int\frac{ log(x) }{x \: dx}$

Using $\int \frac{1}{x} \: dx = ln( |x| )$ We have to evaluate the integral,

We get,

$ln( |x| ) + \int\frac{ log(x) }{x \: dx}$

Now, we have to evaluate the indefinite integral,

$= ln( |x| ) + \frac{ ln( {x}^{2} ) }{2 ln(10) }$

Now, we have to add the constant of integration,

We get,

$= ln( |x| ) + \frac{ ln( {x}^{2} ) }{2 ln(10) } + C ,C \in \R$

$\bf\large\underline\blue{Answer:-}$

$ln( |x| ) + \frac{ ln( {x}^{2} ) }{2 ln(10) } + C ,C \in \R$