Integrate the function \[ f(t)=\frac{t^{10}-\sqrt{t}}{t^{3}} \]

We want to integrate   ∫ [(t¹⁰ - √t) / t³] dt. Step 1. Simplify the integrand. Write the function as follows:   (t¹⁰ - √t) / t³ = t¹⁰/t³ - √t/t³. Recall that t¹⁰/t³ = t^(10 - 3) = t⁷. Also, express √t as t^(1/2); thus   √t/t³ = t^(1/2)/t³ = t^(1/2 - 3) = t^(-5/2). So the integrand becomes:   t⁷ - t^(-5/2). Step 2. Integrate term by term. The integral now is:   ∫ t⁷ dt - ∫ t^(-5/2) dt. For the first term:   ∫ t⁷ dt = t^(7+1)/(7+1) = t⁸/8. For the second term, remember the power rule for integration:   ∫ t^p dt = t^(p+1)/(p+1), provided p ≠ -1. Here, p = -5/2 so that   ∫ t^(-5/2) dt = t^((-5/2)+1)/((-5/2)+1)            = t^(-3/2)/(-3/2) = - (2/3) t^(-3/2). Now, because the term in the original integrand is -t^(-5/2), its integral becomes:   - ∫ t^(-5/2) dt = - [ - (2/3) t^(-3/2) ] = (2/3) t^(-3/2). Step 3. Combine the results. Thus, the integrated function is:   (t⁸/8) + (2/3) t^(-3/2) + C, where C is the constant of integration. Final Answer:   ∫ [(t¹⁰ - √t)/t³] dt = t⁸/8 + (2/3) t^(-3/2) + C.