Answer :
The indefinite integral of [tex]\( \int \frac{108x^8 + 9x^7 + 48x^2}{x^7} \, dx \)[/tex] is [tex]\[ 54x^2 + 9x - 12x^{-4} + C \][/tex], obtained by simplifying the integrand and integrating term by term.
To find the indefinite integral of the given function [tex]\( \int \frac{108x^8 + 9x^7 + 48x^2}{x^7} \, dx \)[/tex], we'll go through the integration process in detail.
First, we can simplify the integrand by dividing each term by [tex]\( x^7 \)[/tex], which gives us:
[tex]\[ \int \frac{108x^8}{x^7} + \frac{9x^7}{x^7} + \frac{48x^2}{x^7} \, dx \][/tex]
This simplifies to:
[tex]\[ \int 108x + 9 + \frac{48}{x^5} \, dx \][/tex]
Now, we can integrate each term separately.
The integral of 108x with respect to x is straightforward:
[tex]\[ \int 108x \, dx = \frac{108x^2}{2} + C_1 = 54x^2 + C_1 \][/tex]
where [tex]\( C_1 \)[/tex]is a constant of integration.
Next, the integral of the constant 9 with respect to x is:
[tex]\[ \int 9 \, dx = 9x + C_2 \][/tex]
where [tex]\( C_2 \)[/tex] is another constant of integration.
Finally, the integral of [tex]\( \frac{48}{x^5} \)[/tex] with respect to x is found by integrating a power of x:
[tex]\[ \int \frac{48}{x^5} \, dx = 48 \int x^{-5} \, dx = 48 \left( \frac{x^{-5 + 1}}{-5 + 1} \right) + C_3 = 48 \left( \frac{x^{-4}}{-4} \right) + C_3 \][/tex]
[tex]\[ = -12x^{-4} + C_3 \][/tex]
where [tex]\( C_3 \)[/tex] is yet another constant of integration.
Combining all these results, the integral of the entire function is:
[tex]\[ 54x^2 + 9x - 12x^{-4} + C \][/tex]
where [tex]\( C = C_1 + C_2 + C_3 \)[/tex] is the overall constant of integration.
Thus, the indefinite integral of the function [tex]\( \int \frac{108x^8 + 9x^7 + 48x^2}{x^7} \, dx \)[/tex] is:
[tex]\[ 54x^2 + 9x - 12x^{-4} + C \][/tex]
This final expression gives us the antiderivative of the original integrand, and C signifies that there are an infinite number of antiderivatives, differing only by a constant.
