How do you find the indefinite integral of #int (x(x-2))/(x-1)^3#?

Question

Luke Alvarez · Accepted Answer

Substitute #u=x-1# to get #ln|x-1|+1/(2(x-1)^2)+C#

#int (x(x-2))/(x-1)^3dx# Substitute #u=x-1#, #dx=du#: #=int((u+1)(u-1))/u^3du# #=int u^-1-u^-3dx# #=ln|u|+1/2u^-2+C# #=ln|x-1|+1/(2(x-1)^2)+C#

Whenever the denominator of the rational function is just a (positive) power of a (linear?) expression this will work, because it "pushes all the messy stuff upstairs".

Samantha Adkison · Answer

undefined To find the indefinite integral of $ \int \frac{x(x-2)}{(x-1)^3} \, dx $:

1. Perform partial fraction decomposition:
$$ \frac{x(x-2)}{(x-1)^3} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{(x-1)^3} $$

2. Solve for $ A $, $ B $, and $ C $ by equating coefficients.

3. Integrate each term separately.

4. The indefinite integral of $ \int \frac{x(x-2)}{(x-1)^3} \, dx $ will be the sum of the integrals obtained from each term.

Ezra Adams · Answer

undefined To find the indefinite integral of $\int \frac{x(x-2)}{(x-1)^3} \, dx$, you can use partial fraction decomposition followed by integration by substitution. Here are the steps:

1. Perform partial fraction decomposition to express the integrand as a sum of simpler fractions.
2. Integrate each of the simpler fractions obtained from the partial fraction decomposition.
3. Combine the results to find the indefinite integral of the original expression.

If you need further clarification on any of these steps or have any other questions, feel free to ask!