integral of (x-2)/(x^3) — step-by-step solution

1) Let's integrate:

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

You can decompose into partial fractions:

\frac{x - 2}{x^{3}} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{3}}

To find the remaining unknowns, multiply both sides by the denominator:

x^{3}

, which yields:

x - 2 = A x^{2} + B x + C

2) By identifying the different terms, you obtain the system of equations:

[1]:

0 = A

[2]:

1 = B

[3]:

-2 = C

Solution obtained [1]:

A = 0

3) Solving for B:

1 = B

[2]

Solution found [2]:

B = 1

4) Let's solve for C:

-2 = C

[3]

Solution obtained [3]:

C = -2

5) Finally, substitute the unknowns back into the initial partial fraction decomposition:

\frac{0}{x} + \frac{1}{x^{2}} - \frac{2}{x^{3}}

0 + \frac{1}{x^{2}} - \frac{2}{x^{3}}

\frac{1}{x^{2}} - \frac{2}{x^{3}}

6) Let's integrate:

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

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

For the math expression:

\frac{1}{x^{2}}

, we know the derivative formula:

\frac{d}{d x} (a^{n}) = n a^{(n - 1)}

, so we try differentiating with ^(m+1)

7) Let's differentiate:

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

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

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

(where f =

x

)

-1 \cdot x^{-2}

- x^{-2}

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

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

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

For the math expression:

\frac{2}{x^{3}}

, we know the derivative formula:

\frac{d}{d x} (a^{n}) = n a^{(n - 1)}

, so we try differentiating with ^(m+1)

8) Let's differentiate:

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

\frac{d}{d x} (x^{-2})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

(where f =

x

)

-2 x^{-3}

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

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

- 1 \cdot \frac{1}{x} - (\frac{2}{-2} \cdot \frac{1}{x^{2}})

- \frac{1}{x} - (- \frac{2}{2} \cdot \frac{1}{x^{2}})

ℹ

- \frac{2}{2}

-1

‖

- (-1 \cdot \frac{1}{x^{2}})

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

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

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

Indefinite integrals are defined up to an additive constant C, so this yields:

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

▶Antiderivative:

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

, where C is a constant