Evaluate the integral: I = integral (1 - 1/(x²)) sqrt(x
Evaluate the integral: $$ I = \int \left(1 - \frac{1}{x^2}\right) \sqrt{x\sqrt{x}} \, dx. $$
Explanation of Method
A nested radical such as $\sqrt{x\sqrt{x}}$ collapses to a single rational power of $x$ once the inner root is written as a fractional exponent. After that simplification, distributing the remaining factor over the parentheses reduces the integral to a sum of pure power terms, each handled by the power rule.
- Rewrite the nested radical as a single power of $x$.
- Distribute the factor over the parenthesized expression.
- Integrate each resulting power of $x$ term by term.
- Combine the antiderivatives and add the constant of integration.
Step 1 · Simplify the integrand
The term $\sqrt{x\sqrt{x}}$ can be rewritten as:
$$ \sqrt{x\sqrt{x}} = \sqrt{x \cdot x^{1/2}} = \sqrt{x^{3/2}} = x^{3/4}. $$
Substituting this back into the integral, the expression becomes:
$$ I = \int \left(1 - \frac{1}{x^2}\right) x^{3/4}\,dx. $$
Now expand the terms:
$$ \int \left(1 - \frac{1}{x^2}\right) x^{3/4}\,dx = \int x^{3/4}\,dx - \int \frac{x^{3/4}}{x^2}\,dx. $$
Simplify each term. The first term remains $\displaystyle \int x^{3/4}\,dx$. The second term simplifies as:
$$ \frac{x^{3/4}}{x^2} = x^{3/4-2} = x^{-5/4}. $$
Thus, the integral becomes:
$$ I = \int x^{3/4}\,dx - \int x^{-5/4}\,dx. $$
Step 2 · Integrate each term
We use the power rule for integration:
$$ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1. $$
First term: $\displaystyle \int x^{3/4}\,dx$:
$$ \int x^{3/4}\,dx = \frac{x^{3/4+1}}{3/4+1} = \frac{x^{7/4}}{7/4} = \frac{4}{7}x^{7/4}. $$
Second term: $\displaystyle \int x^{-5/4}\,dx$:
$$ \int x^{-5/4}\,dx = \frac{x^{-5/4+1}}{-5/4+1} = \frac{x^{-1/4}}{-1/4} = -4x^{-1/4}. $$
Step 3 · Combine the results
Combine the results of both integrals:
$$ I = \frac{4}{7}x^{7/4} - \left(-4x^{-1/4}\right) + C = \frac{4}{7}x^{7/4} + 4x^{-1/4} + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Rewritten as | Antiderivative |
|---|---|---|
| $\sqrt{x\sqrt{x}}$ | $x^{3/4}$ | $\dfrac{4}{7}x^{7/4}$ |
| $x^{3/4}/x^2$ | $x^{-5/4}$ | $-4x^{-1/4}$ (subtracted, giving $+4x^{-1/4}$) |
Additional Notes
- The domain of $I$ requires $x > 0$, since the nested square root and the negative power $x^{-2}$ both need $x \neq 0$ and non-negative $x$ for real values.
- Nested radicals like $\sqrt{x\sqrt{x}}$ always reduce to a single power of $x$: here $x \cdot x^{1/2} = x^{3/2}$, and the outer square root halves the exponent to $3/4$.
- The result can be checked by differentiating $\dfrac{4}{7}x^{7/4} + 4x^{-1/4}$ and confirming it returns the original integrand.
Result
The solution to the integral is: