91Ó°ÊÓ

To integrate the function, we break it down into individual terms and apply basic integration rules for each term. $$\int\left(3 x^{1 / 3}+4 x^{-1 / 3}+6\right) d x =\int 3 x^{1 / 3} d x + \int 4 x^{-1 / 3} d x + \int 6 d x$$ Now integrate each term separately. For the first term: $$\int 3x^{1/3} dx$$ We will use the power rule, which states that: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ Applying the power rule for the term: $$\int 3x^{1/3} dx = 3\frac{x^{(1/3)+1}}{(1/3)+1} + C_1 = \frac{9x^{4/3}}{4} + C_1$$ For the second term: $$\int 4x^{-1/3} dx$$ Applying the power rule again: $$\int 4x^{-1/3} dx = 4\frac{x^{(-1/3)+1}}{(-1/3)+1} + C_2 = 6x^{2/3} + C_2$$ For the third term: $$\int 6 dx$$ This is a constant, so the integral is simply 6x plus a constant: $$\int 6 dx = 6x + C_3$$ Now combine the results of the three integrals: $$\int\left(3 x^{1 / 3}+4 x^{-1 / 3}+6\right) d x = \frac{9x^{4/3}}{4} + 6x^{2/3} + 6x + C_1 + C_2 + C_3$$ Combine the constants into a single constant, C: $$\int\left(3 x^{1 / 3}+4 x^{-1 / 3}+6\right) d x = \frac{9x^{4/3}}{4} + 6x^{2/3} + 6x + C$$

We will now differentiate the result to see if it matches the original function. Differentiate: $$\frac{d}{dx} \left(\frac{9x^{4/3}}{4} + 6x^{2/3} + 6x + C\right)$$ Apply the power rule for differentiation: $$(\frac{d}{dx} x^n = nx^{(n-1)})$$ For the first term: $$\frac{d}{dx}\left(\frac{9x^{4/3}}{4}\right) = 3x^{(4/3)-1} = 3x^{1/3}$$ For the second term: $$\frac{d}{dx}\left(6x^{2/3}\right) = \frac{4}{x^{1/3}} = 4x^{-1/3}$$ For the third term: $$\frac{d}{dx} (6x) = 6$$ For the constant: $$\frac{d}{dx}(C) = 0$$ Now, add the derivatives of the three terms: $$\frac{d}{dx} \left(\frac{9x^{4/3}}{4} + 6x^{2/3} + 6x + C\right) = 3x^{1/3} + 4x^{-1/3} + 6$$ The derivative of our integral result matches the original function, so we have found the correct indefinite integral: $$\int\left(3 x^{1 / 3}+4 x^{-1 / 3}+6\right) d x = \frac{9x^{4/3}}{4} + 6x^{2/3} + 6x + C$$