Factor the difference of squares $x^2-1$ as the product of two conjugated binomials
Rewrite the fraction $\frac{x}{\left(x+1\right)\left(x-1\right)}$ in $2$ simpler fractions using partial fraction decomposition
Expand the integral $\int\left(\frac{1}{2\left(x+1\right)}+\frac{1}{2\left(x-1\right)}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{1}{2\left(x+1\right)}dx$ 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 $x+1$ 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
Substituting $u$ and $dx$ in the integral and simplify
The integral $\frac{1}{2}\int\frac{1}{u}du$ results in: $\frac{1}{2}\ln\left(x+1\right)$
The integral $\int\frac{1}{2\left(x-1\right)}dx$ results in: $\frac{1}{2}\ln\left(x-1\right)$
Gather the results of all integrals
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!
