Evaluate the integral: integral (x dx)/(5 + x^4)
Evaluate the integral: $$ \int \frac{x\,dx}{5 + x^4}. $$
Explanation of Method
Since $x^4=(x^2)^2$, substituting $u=x^2$ turns the denominator into $5+u^2$, a standard arctangent form, while $x\,dx$ converts neatly into $\tfrac12 du$.
- Substitute $u=x^2$ to expose the arctangent structure.
- Convert $x\,dx$ into a constant multiple of $du$.
- Apply the standard arctangent integral formula.
- Substitute back and add the constant of integration.
Step 1 · Substitution
We solve this integral using the substitution method. Let:
$$ u = x^2. $$
Then, differentiate both sides with respect to $x$:
$$ \frac{du}{dx} = 2x, \quad \text{so} \quad du = 2x\,dx. $$
Therefore, $x\,dx = \dfrac{du}{2}$.
Step 2 · Substitute into the integral
Substitute this into the original integral:
$$ \int \frac{x\,dx}{5+x^4} = \int \frac{\frac{du}{2}}{5+u^2} = \frac{1}{2}\int \frac{du}{5+u^2}. $$
Step 3 · Apply the standard result
We recognize that the integral $\displaystyle \int \frac{du}{a^2+u^2}$ is a standard integral, whose solution is:
$$ \frac{1}{a}\arctan\left(\frac{u}{a}\right). $$
In our case, $a^2=5$, so $a=\sqrt5$. Thus, we have:
$$ \frac{1}{2}\int \frac{du}{5+u^2} = \frac{1}{2}\cdot \frac{1}{\sqrt5}\arctan\left(\frac{u}{\sqrt5}\right). $$
Step 4 · Substitute back
Now, substitute $u=x^2$ back into the solution:
$$ \frac{1}{2\sqrt{5}}\arctan\left(\frac{x^2}{\sqrt{5}}\right) + C. $$
Summary Table
| Quantity | Value |
|---|---|
| Substitution | $u = x^2$ |
| $x\,dx$ in terms of $du$ | $\tfrac12 du$ |
| Antiderivative | $\tfrac{1}{2\sqrt5}\arctan(u/\sqrt5)$ |
Additional Notes
- The integrand is defined for all real $x$, since $5+x^4 \geq 5 > 0$ always.
- Recognizing $x^4$ as $(x^2)^2$ is the key first step — it reveals the hidden arctangent structure in what otherwise looks like a quartic denominator.
- The result can be checked by differentiating $\tfrac{1}{2\sqrt5}\arctan(x^2/\sqrt5)$ with the chain rule and confirming it returns $\dfrac{x}{5+x^4}$.
Result
The solution to the integral is: