Certainly! Let’s find the indefinite integral of the function [tex]\(\int \left(10x^4 - 4x + \frac{3}{x^4}\right) \, dx\)[/tex].
First, we will break down the integral into separate parts:
[tex]\[
\int \left(10x^4 - 4x + \frac{3}{x^4}\right) \, dx = \int 10x^4 \, dx - \int 4x \, dx + \int \frac{3}{x^4} \, dx.
\][/tex]
Now, we will integrate each term one by one:
1. Integrating [tex]\(10x^4\)[/tex]:
[tex]\[
\int 10x^4 \, dx
\][/tex]
Using the power rule for integration [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], we get:
[tex]\[
\int 10x^4 \, dx = 10 \cdot \frac{x^{4+1}}{4+1} = 10 \cdot \frac{x^5}{5} = 2x^5.
\][/tex]
2. Integrating [tex]\(-4x\)[/tex]:
[tex]\[
\int -4x \, dx
\][/tex]
Again, using the power rule for integration:
[tex]\[
\int -4x \, dx = -4 \cdot \frac{x^{1+1}}{1+1} = -4 \cdot \frac{x^2}{2} = -2x^2.
\][/tex]
3. Integrating [tex]\(\frac{3}{x^4}\)[/tex]:
[tex]\[
\int \frac{3}{x^4} \, dx
\][/tex]
We can rewrite [tex]\(\frac{3}{x^4}\)[/tex] as [tex]\(3x^{-4}\)[/tex], and then use the power rule for integration:
[tex]\[
\int 3x^{-4} \, dx = 3 \cdot \frac{x^{-4+1}}{-4+1} = 3 \cdot \frac{x^{-3}}{-3} = -\frac{3}{3}x^{-3} = -x^{-3}.
\][/tex]
Rewriting [tex]\(x^{-3}\)[/tex] back as [tex]\(\frac{1}{x^3}\)[/tex], we have:
[tex]\[
\int \frac{3}{x^4} \, dx = -\frac{1}{x^3}.
\][/tex]
Now, we combine all these results:
[tex]\[
\int \left(10x^4 - 4x + \frac{3}{x^4}\right) \, dx = 2x^5 - 2x^2 - \frac{1}{x^3} + C,
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
Therefore, the indefinite integral of [tex]\(\int \left(10x^4 - 4x + \frac{3}{x^4}\right) \, dx\)[/tex] is:
[tex]\[
2x^5 - 2x^2 - \frac{1}{x^3} + C.
\][/tex]
