Check the solution to "Evaluate the following integrals. ∫dxx3−x2"

Tazmin Horton

Answered question

2021-01-06

Evaluate the following integrals.

\int \frac{d x}{x^{3} - x^{2}}

Jaylen Fountain

Skilled2021-01-07Added 169 answers

Step 1
Consider the provided integral,

\int \frac{d x}{x^{3} - x^{2}}

Evaluate the following integrals.
We can write as,

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

First, we find the partial fraction.

\frac{1}{x^{2} (x - 1)} = \frac{a_{0}}{x} + \frac{a_{1}}{x^{2}} + \frac{a_{3}}{x - 1}

...(1)
Step 2
Now, multiply the equation by the provided denominator.

\frac{1 \cdot x^{2} (x - 1)}{x^{2} (x - 1)} = \frac{a_{0} x^{2} (x - 1)}{x} + \frac{a_{1} x^{2} (x - 1)}{x^{2}} + \frac{a_{2} x^{2} (x - 1)}{x} - 1

1 = a_{0} x (x - 1) + a_{1} (x - 1) + a_{2} x^{2}

Compare the both the sides of the coefficient.

a_{1} = - 1, a_{2} = 1

Using above values find the

a_{0}

a_{0} = - 1

.
Step 3
Put the above values in the equation (1).

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

= - \frac{1}{x} = \frac{1}{x^{2}} + \frac{1}{x - 1}

So, the integrals becomes.

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

= - \int \frac{1}{x} d x - \int \frac{1}{x^{2}} d x + \int \frac{1}{x - 1} d x

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

= - \ln | x | + \frac{1}{x} + \ln | x - 1 | + C

Hence.

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