Questions: Evaluate the indefinite integral given below. [ intleft(9 x^1/3+8 x^1/4right) d x ]

Solution Steps

To evaluate the indefinite integral, we need to integrate each term separately. We will use the power rule for integration, which states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), provided \(n \neq -1\).

1. Identify the terms to be integrated: \(9x^{\frac{1}{3}}\) and \(8x^{\frac{1}{4}}\).
2. Apply the power rule to each term.
3. Combine the results and add the constant of integration \(C\).

We need to evaluate the indefinite integral: \[ \int \left(9 x^{\frac{1}{3}} + 8 x^{\frac{1}{4}}\right) d x \]

Using the power rule for integration, we integrate each term separately:

1. For \(9 x^{\frac{1}{3}}\): \[ \int 9 x^{\frac{1}{3}} d x = 9 \cdot \frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} = 9 \cdot \frac{x^{\frac{4}{3}}}{\frac{4}{3}} = 6.75 x^{\frac{4}{3}} \]
2. For \(8 x^{\frac{1}{4}}\): \[ \int 8 x^{\frac{1}{4}} d x = 8 \cdot \frac{x^{\frac{1}{4} + 1}}{\frac{1}{4} + 1} = 8 \cdot \frac{x^{\frac{5}{4}}}{\frac{5}{4}} = 6.4 x^{\frac{5}{4}} \]

Combining the results from both integrals, we have: \[ \int \left(9 x^{\frac{1}{3}} + 8 x^{\frac{1}{4}}\right) d x = 6.75 x^{\frac{4}{3}} + 6.4 x^{\frac{5}{4}} + C \]

Final Answer