To solve the integral:
[tex]\[
\int \frac{x^3}{x^4-4} \, dx
\][/tex]
we use the given substitution [tex]\( u = x^4 - 4 \)[/tex].
1. Find [tex]\( du \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( dx \)[/tex]:
We start by differentiating [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
u = x^4 - 4
\][/tex]
[tex]\[
\frac{du}{dx} = 4x^3
\][/tex]
Therefore,
[tex]\[
du = 4x^3 \, dx
\][/tex]
Rearranging to solve for [tex]\( dx \)[/tex],
[tex]\[
dx = \frac{du}{4x^3}
\][/tex]
2. Rewrite the integral in terms of [tex]\( u \)[/tex]:
Substitute [tex]\( u = x^4 - 4 \)[/tex] and [tex]\( dx = \frac{du}{4x^3} \)[/tex] into the original integral:
[tex]\[
\int \frac{x^3}{x^4 - 4} \, dx = \int \frac{x^3}{u} \cdot \frac{du}{4x^3}
\][/tex]
Simplify the integral:
[tex]\[
\int \frac{1}{u} \cdot \frac{du}{4} = \frac{1}{4} \int \frac{1}{u} \, du
\][/tex]
3. Evaluate the integral:
The integral [tex]\( \int \frac{1}{u} \, du \)[/tex] is a standard integral:
[tex]\[
\int \frac{1}{u} \, du = \ln|u| + C
\][/tex]
Therefore,
[tex]\[
\frac{1}{4} \int \frac{1}{u} \, du = \frac{1}{4} \ln|u| + C
\][/tex]
Substitute back [tex]\( u = x^4 - 4 \)[/tex]:
[tex]\[
\frac{1}{4} \ln|x^4 - 4| + C
\][/tex]
Thus, the evaluated integral is:
[tex]\[
\int \frac{x^3}{x^4 - 4} \, dx = \frac{1}{4} \ln|x^4 - 4| + C
\][/tex]
