To solve the integral [tex]\(\int \frac{2x}{1+x^4} \, dx\)[/tex], we need to recognize that it can be approached by using a substitution that simplifies the integrand. Here are the detailed steps to solve it:
1. Substitution:
Let [tex]\(u = x^2\)[/tex]. Then, [tex]\(du = 2x \, dx\)[/tex]. Notice how the differential [tex]\(2x \, dx\)[/tex] appears in the numerator of our integrand.
2. Rewriting the Integral:
Substitute [tex]\(u = x^2\)[/tex] and [tex]\(du = 2x \, dx\)[/tex] into the given integral:
[tex]\[
\int \frac{2x}{1+x^4} \, dx = \int \frac{1}{1+u^2} \, du
\][/tex]
Here, we have utilized the substitution [tex]\(u = x^2\)[/tex], hence [tex]\(x^4 = (x^2)^2 = u^2\)[/tex].
3. Recognizing the Integral of a Standard Form:
The integral [tex]\(\int \frac{1}{1+u^2} \, du\)[/tex] is a standard integral form. Recall that:
[tex]\[
\int \frac{1}{1+u^2} \, du = \arctan(u) + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
4. Substitute Back the Variable [tex]\(u\)[/tex]:
Since [tex]\(u = x^2\)[/tex], we substitute back to get:
[tex]\[
\arctan(u) = \arctan(x^2)
\][/tex]
5. Final Answer:
Therefore, the integral [tex]\(\int \frac{2x}{1+x^4} \, dx\)[/tex] evaluates to:
[tex]\[
\arctan(x^2) + C
\][/tex]
So, the final answer is:
[tex]\[
\int \frac{2x}{1+x^4} \, dx = \arctan(x^2) + C
\][/tex]
