Immediate integrals

Calculate the following indefinite integral of a basic polynomial function using the properties of linearity (separating sums and taking out constants):

$$\int (4x^3 - 6x^2 + 2x - 5) \, dx$$

Calculate the indefinite integral of the following polynomial function containing terms with fractional and negative powers. Remember to express the result in the most simplified form possible:

$$\int \left( 4x^3 - \frac{3}{x^2} + \sqrt[3]{x} \right) dx$$

To integrate the following rational function, you must first simplify the expression by dividing each term in the numerator by the denominator:

$$\int \frac{5x^4 - 2x^2 + 3}{x^2} \, dx$$

Calculate the following indefinite integral using the properties of powers and the rules for standard integrals:

$$\int \frac{3x - \sqrt{x^3} + 2}{x^2} \, dx$$

Next, find the antiderivative that takes the value $4$ when $x=1$.

Find the antiderivative of the function $f(x) = \frac{1}{x} + \cos(x)$. Remember which function has a derivative of $\frac{1}{x}$ and pay attention to the modulus:

$$\int (\frac{1}{x} + \cos x) \, dx$$

Determine the value of the following near-immediate integrals (logarithmic and exponential types with constants):

1. a) $\int \frac{3}{x+2} \, dx$

1. b) $\int e^{5x} \, dx$

A market study indicates that the growth rate of followers for a new influencer is given by the function $f(x) = 6x + 10$ (where $x$ is the time in days).

Find the function $F(x)$ that gives the total number of followers, knowing that at the time of launch (day $x=0$) they already had $500$ followers.

In a mobile phone case manufacturing company, the marginal cost (the cost of producing an additional unit) is estimated using the function $C'(x) = 0,6x + 4$.

Obtain the expression for total cost $C(x)$ if it is known that the company's fixed costs (rent, electricity, etc.), which correspond to producing 0 units, amount to $1200$ euros.

Calculate the indefinite integral of this function that combines an exponential with base $e$ and a negative power. Be careful with the sign when applying the power rule:

$$\int (e^x - \frac{2}{x^3}) \, dx$$