Evaluate the integral: I = integral dx/((5x - 2)^(5/2))
Evaluate the integral: $$ I = \int \frac{dx}{(5x - 2)^{5/2}}. $$
Explanation of Method
The integrand $(5x-2)^{-5/2}$ is again a power of a linear expression, this time with a negative fractional exponent. The linear substitution $u=5x-2$ converts it to a routine power-rule integral in $u$, after which the result is written back in terms of $x$.
- Substitute $u$ for the linear expression.
- Convert $dx$ into $du$ using the constant derivative.
- Integrate the resulting negative power using the power rule.
- Substitute back and add the constant of integration.
Step 1 · Use substitution
To simplify the integral, we will use substitution. Let:
$$ u = 5x - 2. $$
Then, differentiate $u$ with respect to $x$:
$$ \frac{du}{dx} = 5 \quad \Rightarrow \quad dx = \frac{du}{5}. $$
Substitute this into the integral:
$$ I = \int \frac{1}{u^{5/2}}\cdot \frac{du}{5} = \frac{1}{5}\int u^{-5/2}\,du. $$
Step 2 · Integrate
We now integrate $u^{-5/2}$. Using the power rule for integration:
$$ \int u^n\,du = \frac{u^{n+1}}{n+1} + C \quad \text{for } n\neq -1, $$
we get:
$$ \int u^{-5/2}\,du = \frac{u^{-3/2}}{-3/2} = -\frac{2}{3}u^{-3/2}. $$
Step 3 · Substitute back $u = 5x - 2$
Now, substitute back $u = 5x-2$ into the result:
$$ I = \frac{1}{5}\cdot \left(-\frac{2}{3}(5x-2)^{-3/2}\right). $$
Step 4 · Add the constant of integration
Finally, we add the constant of integration $C$ to the result:
$$ I = -\frac{2}{15}(5x-2)^{-3/2} + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Substitution | $u = 5x-2$ |
| $dx$ in terms of $du$ | $\tfrac{1}{5}du$ |
| Antiderivative in $u$ | $-\tfrac{2}{3}u^{-3/2}$ |
Additional Notes
- The domain of $I$ requires $x > \tfrac{2}{5}$, since $5x-2>0$ is needed for the fractional power (with even-denominator-like behavior) to be real.
- The final coefficient $-\tfrac{2}{15}$ comes from multiplying the substitution factor $\tfrac15$ by the power-rule factor $-\tfrac23$.
- The result can be checked by differentiating $-\tfrac{2}{15}(5x-2)^{-3/2}$ with the chain rule and confirming it returns $(5x-2)^{-5/2}$.
Result
The solution to the integral is: