Evaluate the integral: integral dx/((x-2)³)
Evaluate the integral: $$ \int \frac{dx}{(x-2)^3}. $$
Explanation of Method
A reciprocal power of a linear expression, $\frac{1}{(x-a)^n}$, is integrated by substituting $u = x-a$, which has derivative $1$, so $dx = du$ directly. The resulting integral $\int u^{-n}\,du$ is a routine power-rule application.
- Substitute $u$ for the linear expression.
- Rewrite the integral as a negative power of $u$.
- Integrate using the power rule.
- Substitute back and add the constant of integration.
Step 1 · Use substitution
To simplify the integral, we will use the substitution method. Let:
$$ u = x - 2. $$
Then, the derivative of $u$ with respect to $x$ is:
$$ \frac{du}{dx} = 1 \quad \Rightarrow \quad dx = du. $$
Substituting into the integral, we get:
$$ I = \int \frac{du}{u^3}. $$
Step 2 · Integrate
Now, we integrate $\dfrac{1}{u^3}$. Using the power rule for integration, which states that:
$$ \int u^n\,du = \frac{u^{n+1}}{n+1} \quad \text{for } n\neq -1, $$
we can integrate $u^{-3}$ as follows:
$$ \int u^{-3}\,du = \frac{u^{-2}}{-2} = -\frac{1}{2u^2}. $$
Step 3 · Substitute back $u = x - 2$
Now, we substitute back $u = x-2$ into the result:
$$ -\frac{1}{2u^2} = -\frac{1}{2(x-2)^2}. $$
Step 4 · Add the constant of integration
Finally, we add the constant of integration $C$ to the result:
$$ I = -\frac{1}{2(x-2)^2} + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Substitution | $u = x-2$ |
| $dx$ in terms of $du$ | $du$ |
| Antiderivative in $u$ | $-\tfrac{1}{2}u^{-2}$ |
Additional Notes
- The domain of $I$ requires $x \neq 2$, since the original integrand is undefined there.
- Because the derivative of $u=x-2$ is exactly $1$, this substitution introduces no extra constant factor, unlike Problem 23 where the coefficient $-2$ inside $u$ produced a $-\tfrac12$ factor.
- The result can be checked by differentiating $-\dfrac{1}{2(x-2)^2}$ and confirming it returns $\dfrac{1}{(x-2)^3}$.
Result
The solution to the integral is: