We can solve the integral $\int\frac{1}{\left(5x-1\right)\ln\left(5x-1\right)}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $5x-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
We can solve the integral $\int\frac{1}{u\ln\left(u\right)}du$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $v$), which when substituted makes the integral easier. We see that $\ln\left(u\right)$ it's a good candidate for substitution. Let's define a variable $v$ and assign it to the choosen part
Now, in order to rewrite $du$ in terms of $dv$, we need to find the derivative of $v$. We need to calculate $dv$, we can do that by finding the derivative of the equation above
Isolate $du$ in the previous equation
Substituting $v$ and $du$ in the integral and simplify
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
Replace $v$ with the value that we assigned to it in the beginning: $\ln\left(u\right)$
Replace $u$ with the value that we assigned to it in the beginning: $5x-1$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Try other ways to solve this exercise
Get a preview of step-by-step solutions.
Earn solution credits, which you can redeem for complete step-by-step solutions.
Save your favorite problems.
Become premium to access unlimited solutions, download solutions, discounts and more!
