Questions: Calcule as integrais indefinidas: (a) ∫(x+3) dx

Calculate the indefinite integral: $\int(x+3) dx$

Identify the terms in the integrand

The integrand is $(x+3)$, which can be separated into two terms: $x$ and $3$.

Apply the linearity property of integration

Using the property that $\int(f(x) + g(x))dx = \int f(x)dx + \int g(x)dx$, we can write: $\int(x+3)dx = \int x dx + \int 3 dx$

Evaluate each integral separately

For $\int x dx$, we use the power rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$

$\int x dx = \int x^1 dx = \frac{x^2}{2} + C_1$

For $\int 3 dx$, we integrate the constant: $\int 3 dx = 3x + C_2$

Combine the results

$\int(x+3)dx = \frac{x^2}{2} + 3x + C$

Where $C = C_1 + C_2$ is the arbitrary constant of integration.

$\boxed{\int(x+3)dx = \frac{x^2}{2} + 3x + C}$

$\boxed{\int(x+3)dx = \frac{x^2}{2} + 3x + C}$