Separate into Two Integrals: We are given the integral of a sum of two functions, which can be separated into two integrals. The integral of x with respect to x, and the integral of 2 with respect to x. We can use the linearity of the integral to separate them. Calculation: ∫(x dx) + ∫(2 dx)Integrate x with Power Rule: To integrate x with respect to x, we use the power rule for integration. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. In our case, n = 1. Calculation: ∫x dx = (x^(1+1))/(1+1) + C = (x^2)/2 + CIntegrate Constant 2: To integrate the constant 2 with respect to x, we use the fact that the integral of a constant a with respect to x is ax + C. Calculation: ∫2 dx = 2x + CCombine Results for Final Answer: Now we combine the results of the two integrals to get the final answer. Calculation: (x^2)/2 + 2x + C

Calculus

Find indefinite integrals using the substitution and by parts

Full solution

\int x d x+\int 2 d x

Separate into Two Integrals: We are given the integral of a sum of two functions, which can be separated into two integrals. The integral of $x$ with respect to $x$ , and the integral of $2$ with respect to $x$ . We can use the linearity of the integral to separate them. $\newline$ Calculation: $\int(x \, dx) + \int(2 \, dx)$
Integrate $x$ with Power Rule: To integrate $x$ with respect to $x$ , we use the power rule for integration. The power rule states that $\int x^n \, dx = \frac{x^{(n+1)}}{n+1} + C$ , where $n \neq -1$ . In our case, $n = 1$ . $\newline$ Calculation: $\int x \, dx = \frac{x^{(1+1)}}{1+1} + C = \frac{x^2}{2} + C$
Integrate Constant $2$ : To integrate the constant $2$ with respect to $x$ , we use the fact that the integral of a constant $a$ with respect to $x$ is $ax + C$ .
Calculation: $\int 2 \, dx = 2x + C$
Combine Results for Final Answer: Now we combine the results of the two integrals to get the final answer. $\newline$ Calculation: $(\frac{x^2}{2} + 2x + C)$