top hat practice\n1. ∫x³ + 3x⁻¹ + 7dx

Explanation:

Step1: Apply sum - rule of integration

$\int(x^{3}+3x^{-1}+7)dx=\int x^{3}dx + 3\int x^{-1}dx+\int7dx$

Step2: Integrate each term

For $\int x^{3}dx$, using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int x^{3}dx=\frac{x^{4}}{4}+C_1$. For $\int x^{-1}dx$, since $\int\frac{1}{x}dx=\ln|x|+C_2$, then $3\int x^{-1}dx = 3\ln|x|+3C_2$. For $\int7dx$, since $\int kdx=kx + C_3$ ($k$ is a constant), then $\int7dx=7x + C_3$.

Step3: Combine the results

$\frac{x^{4}}{4}+3\ln|x|+7x + C$ (where $C = C_1+3C_2 + C_3$)

Answer:

$\frac{x^{4}}{4}+3\ln|x|+7x + C$