To solve the given indefinite integral [tex]\(\int \frac{4 x^7 + 8 x^5}{x^4} \, dx\)[/tex], we will follow these steps:
1. Simplify the integrand:
[tex]\[
\frac{4 x^7 + 8 x^5}{x^4}
\][/tex]
We can simplify the fraction by dividing each term in the numerator by [tex]\(x^4\)[/tex]:
[tex]\[
\frac{4 x^7}{x^4} + \frac{8 x^5}{x^4} = 4 x^{7-4} + 8 x^{5-4} = 4 x^3 + 8 x
\][/tex]
2. Rewrite the integral with the simplified integrand:
[tex]\[
\int (4 x^3 + 8 x) \, dx
\][/tex]
3. Integrate each term separately:
- The integral of [tex]\(4 x^3\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\int 4 x^3 \, dx = 4 \cdot \frac{x^{3+1}}{3+1} = 4 \cdot \frac{x^4}{4} = x^4
\][/tex]
- The integral of [tex]\(8 x\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\int 8 x \, dx = 8 \cdot \frac{x^{1+1}}{1+1} = 8 \cdot \frac{x^2}{2} = 4 x^2
\][/tex]
4. Combine the results and include the constant of integration:
[tex]\[
\int (4 x^3 + 8 x) \, dx = x^4 + 4 x^2 + C
\][/tex]
Thus, the indefinite integral is:
[tex]\[
\int \frac{4 x^7 + 8 x^5}{x^4} \, dx = x^4 + 4 x^2 + C
\][/tex]
