91Ó°ÊÓ

To integrate the function, apply the power rule for integration. The power rule states that if \(n\) is constant and \(r\) is a constant real number, then: $$\int x^r dx = \frac{x^{r+1}}{r+1} + C$$ where C is the constant of integration. Apply the power rule to the given function: $$\int\left(3 x^{5}-5 x^{9}\right) d x = \int(3x^5)dx - \int(5x^9)dx$$

Now, integrate each term separately using the power rule: $$\int(3x^5)dx = 3\int(x^5)dx = 3\frac{x^{5+1}}{5+1}+C_1 = \frac{3x^6}{6}+C_1$$ $$\int(5x^9)dx = 5\int(x^9)dx = 5\frac{x^{9+1}}{9+1}+C_2 = \frac{5x^{10}}{10}+C_2$$ Combine both integrals: $$\int\left(3 x^{5}-5 x^{9}\right) d x = \frac{3x^6}{6}+C_1 - \frac{5x^{10}}{10}+C_2$$

Combine the constants of integration, \(C_1\) and \(C_2\), into a single constant, \(C\): $$\int\left(3 x^{5}-5 x^{9}\right) d x = \frac{3x^6}{6} - \frac{5x^{10}}{10}+C$$

Now differentiate the result to verify that it matches the original function. Apply the power rule for differentiation: $$\frac{d}{dx}\left(\frac{3x^6}{6} - \frac{5x^{10}}{10}+C\right) = \frac{d}{dx}\left(\frac{3x^6}{6}\right) - \frac{d}{dx}\left(\frac{5x^{10}}{10}\right)$$ Differentiate each term separately: $$\frac{d}{dx}\left(\frac{3x^6}{6}\right) = 3x^5$$ $$\frac{d}{dx}\left(\frac{5x^{10}}{10}\right) = 5x^9$$ Combine the derivatives: $$\frac{d}{dx}\left(\frac{3x^6}{6} - \frac{5x^{10}}{10}+C\right) = 3x^5 - 5x^9$$ Since the derivative of the result matches the original function, the integration is correct. Thus, the indefinite integral of the given function is: $$\int\left(3 x^{5}-5 x^{9}\right) d x = \frac{3x^6}{6} - \frac{5x^{10}}{10}+C$$