Questions: Evaluate the indefinite integral given below. [ intleft(-4 x^3/2+3 x^2/3-4 x^2right) d x ] Provide your answer below: [ intleft(-4 x^3/2+3 x^2/3-4 x^2right) d x=square ]

Solution Steps

To evaluate the indefinite integral, we will integrate each term of the polynomial separately. The power rule for integration states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), provided \(n \neq -1\). We will apply this rule to each term in the polynomial and then combine the results.

We need to evaluate the indefinite integral

\[ \int\left(-4 x^{\frac{3}{2}} + 3 x^{\frac{2}{3}} - 4 x^{2}\right) d x. \]

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

1. For \(-4 x^{\frac{3}{2}}\): \[ \int -4 x^{\frac{3}{2}} \, dx = -4 \cdot \frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} = -\frac{4}{\frac{5}{2}} x^{\frac{5}{2}} = -\frac{8}{5} x^{\frac{5}{2}}. \]

2. For \(3 x^{\frac{2}{3}}\): \[ \int 3 x^{\frac{2}{3}} \, dx = 3 \cdot \frac{x^{\frac{2}{3} + 1}}{\frac{2}{3} + 1} = 3 \cdot \frac{x^{\frac{5}{3}}}{\frac{5}{3}} = \frac{9}{5} x^{\frac{5}{3}}. \]

3. For \(-4 x^{2}\): \[ \int -4 x^{2} \, dx = -4 \cdot \frac{x^{2 + 1}}{2 + 1} = -\frac{4}{3} x^{3}. \]

Combining all the integrated terms, we have:

\[ \int\left(-4 x^{\frac{3}{2}} + 3 x^{\frac{2}{3}} - 4 x^{2}\right) d x = -\frac{8}{5} x^{\frac{5}{2}} + \frac{9}{5} x^{\frac{5}{3}} - \frac{4}{3} x^{3} + C, \]

where \(C\) is the constant of integration.

Final Answer