Step 1:

Using our knowledge of common integrals, and the integral power rule we can find the integral as follows:

[latex]\small{\int_{1}^{3}(\cos(x) + \frac{2}{x^4}) dx = (\sin(x) – \frac{2}{3x^3})|_{1}^{3}}[/latex]

Step 2:

Now, we can apply the fundamental theorem of calculus. This states that:

[latex]\large{\int_{a}^{b}f(x) dx = F(b) – F(a)}[/latex]

Thus, we have:

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

[latex]\large{= [\sin(3) – \frac{2}{3(3)^3}] – [\sin(1) – \frac{2}{3(1)^3}]}[/latex]

[latex]\large{ =  [\sin(3) – \frac{2}{81}]\:-\: [\sin(1) – \frac{2}{3}]}[/latex]

[latex]\large{ = \sin(3)-\sin(1) + \frac{56}{81}}[/latex]

[latex]\large{\approx -0.0584}[/latex]

Step 3:

So, our final answer is:

[latex]\large{\int_{1}^{3}(\cos(x) + \frac{2}{x^4})dx \approx -0.0584}[/latex]

Solution Complete!