Answered: "Find the indefinite integral. ∫x(x−2)(x−1)3dx"

Deragz

Answered question

2021-12-10

Find the indefinite integral.

\int \frac{x (x - 2)}{{(x - 1)}^{3}} d x

Cassandra Ramirez

Beginner2021-12-11Added 30 answers

Step 1
Consider the integral

\int \frac{x (x - 2)}{{(x - 1)}^{3}} d x

Step 2
Rewrite the integrals as,

\int \frac{{(x - 1)}^{2} - 1}{{(x - 1)}^{3}} d x

Step 3
Use the substitution in the given integral.
u=x-1 and du=dx
Step 4
Integrate the given function with respect to x.

\int \frac{{(x - 1)}^{2} - 1}{{(x - 1)}^{3}} d x = \int \frac{u^{2} - 1}{u^{3}} d u

= \int (\frac{1}{u} - \frac{1}{u^{3}}) d u

= \int \frac{1}{u} d u - \int \frac{1}{u^{3}} d u

= \ln (u) - (- \frac{1}{2 u^{2}})

= \ln (x - 1) + \frac{1}{2 {(x - 1)}^{2}} + C

Step 5
Thus, the value of the given integral is

\ln (x - 1) + \frac{1}{2 {(x - 1)}^{2}} + C

limacarp4

Beginner2021-12-12Added 39 answers

\int \frac{x (x - 2)}{{(x - 1)}^{3}} d x

\int \frac{1}{x - 1} - \frac{1}{{(x - 1)}^{3}} d x

\int \frac{1}{x - 1} d x - \int \frac{1}{{(x - 1)}^{3}} d x

\ln (| x - 1 |) + \frac{1}{2 {(x - 1)}^{2}}

Solution:

\ln (| x - 1 |) + \frac{1}{2 {(x - 1)}^{2}}

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