1. The problem is to understand and apply the basic integral formulas given: - $\int k \, dx = kx + C$, where $k$ is a constant. - $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$. - $\int \frac{1}{x+k} \, dx = \ln|x+k| + C$. - $\int e^{f(x)} \, dx = \frac{e^{f(x)}}{f'(x)} + C$. 2. Explanation of each formula: - For a constant $k$, integrating with respect to $x$ simply multiplies $k$ by $x$ and adds the constant of integration $C$. - For a power function $x^n$, increase the exponent by 1 and divide by the new exponent, provided $n \neq -1$ (since that case leads to a logarithm). - For the reciprocal of a linear function $x+k$, the integral is the natural logarithm of the absolute value of $x+k$ plus $C$. - For an exponential function with a function $f(x)$ in the exponent, the integral is the exponential divided by the derivative of $f(x)$, plus $C$. 3. These formulas are fundamental tools for solving integrals in calculus and can be applied directly when the integrand matches these forms. Final answer: The four basic integral formulas are as stated and can be used to solve integrals of constants, powers, reciprocals of linear functions, and exponentials with function exponents.