Step 1:

We can first apply the integration to each term as follows:

[latex]\large{\int(3x^4-x^3+2x^{\frac{1}{3}}-x^{-2})dx }[/latex]

[latex]\small{= \int(3x^4)dx – \int(x^3)dx + \int(2x^{\frac{1}{3}})dx – \int(x^{-2})dx}[/latex]

Step 2:

Now, we can use the integral power rule to evaluate each integral individually, as shown:

[latex]\large{\int(3x^4)dx = \frac{3}{5}x^5}[/latex]

[latex]\large{\int(x^3)dx = \frac{1}{4}x^4}[/latex]

[latex]\large{\int(2x^{\frac{1}{3}})dx = \frac{3}{2}x^{\frac{4}{3}} }[/latex]

[latex]\large{\int(x^{-2})dx = -\frac{1}{x} }[/latex]

Step 3:

Now, combining everything, we have:

[latex]\large{\int(3x^4-x^3+2x^{\frac{1}{3}} – x^{-2})dx }[/latex]

[latex]\large{= \frac{3}{5}x^5 \:- \:\frac{1}{4}x^4 + \frac{3}{2}x^{\frac{4}{3}} + \frac{1}{x} + C}[/latex]

Solution Complete!