Evaluate the integral: I = integral (sqrt(x^4 + x^-4 +
Evaluate the integral: $$ I = \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3} \, dx. $$
Explanation of Method
An expression of the form $a^2 + 2ab + b^2 = (a+b)^2$ hiding under a square root can be spotted by matching the middle term. Recognizing $x^4 + x^{-4} + 2$ as $(x^2+x^{-2})^2$ removes the radical entirely, leaving a rational expression that splits into standard power-rule and logarithmic terms.
- Recognize the perfect-square pattern under the radical.
- Remove the square root and simplify the resulting fraction.
- Split the integrand into separate terms.
- Integrate each term and combine, adding the constant of integration.
Step 1 · Simplify the square root
The term inside the square root is:
$$ x^4 + x^{-4} + 2. $$
Notice that $x^4 + x^{-4} + 2$ can be rewritten using the identity $a^2+2ab+b^2=(a+b)^2$ with $a = x^2$ and $b = x^{-2}$:
$$ x^4 + x^{-4} + 2 = \left(x^2 + x^{-2}\right)^2. $$
Thus:
$$ \sqrt{x^4+x^{-4}+2} = \sqrt{\left(x^2+x^{-2}\right)^2} = x^2 + x^{-2}. $$
Substituting this back into the integral, we have:
$$ I = \int \frac{x^2+x^{-2}}{x^3}\,dx. $$
Step 2 · Simplify the integrand
Simplify the terms:
$$ \frac{x^2}{x^3} = \frac{1}{x}, \qquad \frac{x^{-2}}{x^3} = x^{-5}. $$
Thus, the integral becomes:
$$ I = \int \frac{1}{x}\,dx + \int x^{-5}\,dx. $$
Step 3 · Integrate each term
First term: $\displaystyle \int \frac{1}{x}\,dx$:
$$ \int \frac{1}{x}\,dx = \ln|x|. $$
Second term: $\displaystyle \int x^{-5}\,dx$. Using the power rule for integration,
$$ \int x^n\,dx = \frac{x^{n+1}}{n+1}, \quad \text{for } n \neq -1, $$
we have:
$$ \int x^{-5}\,dx = \frac{x^{-5+1}}{-5+1} = \frac{x^{-4}}{-4} = -\frac{1}{4x^4}. $$
Step 4 · Combine the results
Combine the results of the two integrals:
$$ I = \ln|x| - \frac{1}{4x^4} + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Rewritten as | Antiderivative |
|---|---|---|
| $x^2/x^3$ | $1/x$ | $\ln|x|$ |
| $x^{-2}/x^3$ | $x^{-5}$ | $-\dfrac{1}{4x^4}$ |
Additional Notes
- The domain of $I$ requires $x \neq 0$, since $x^{-4}$, $x^3$ in the denominator, and $x^{-5}$ are all undefined at $x=0$.
- Spotting perfect-square patterns like $a^2+2ab+b^2$ under a radical is a common technique that eliminates the square root before integration begins.
- The result can be checked by differentiating $\ln|x| - \dfrac{1}{4x^4}$ and confirming it returns the original integrand.
Result
The solution to the integral is: