Antiderivatives of Rational Functions and Integration by Parts

Find the primitive of the following polynomial function using the basic rules of integration:

$f(x) = 6x^5 - 4x^3 + 3x^2 - 2$

Determine the function $F(x)$ which is a primitive of $f(x) = e^x + 2x$ and satisfies the condition of passing through the point $P(0, 5)$.

Find the integral of the function $f(x)$ which combines powers and roots, by first expressing the roots as fractional indices:

$f(x) = \sqrt{x} + \frac{1}{x^2}$

Find the set of all primitive functions of the following rational function, using the linearity properties of the integral:

Calculate the following logarithmic indefinite integrals, adjusting the constants if necessary:

1. a) $\displaystyle \int \frac{5}{x+3} dx$

1. b) $\displaystyle \int \frac{2x}{x^2 - 1} dx$

Calculate the following indefinite integral by observing that the numerator is, or can be easily transformed into, the derivative of the denominator:

Divide the numerator polynomial by the denominator (or decompose the fraction into several terms) to solve the following integral:

Calculate the primitive of the function $g(x)$ by making a prior adjustment in the numerator to obtain an integer part and a fractional part:

A market study analyzes the growth rate of sales for a new model of electric scooter. The function describing the rate of change of sales (in hundreds of units per month) is given by:

$f(x) = \dfrac{x^2 + 2x + 3}{x + 1}$

where $x$ is the time in months. Calculate the family of total sales functions $F(x)$ by obtaining the antiderivative of this function. (Note: Perform polynomial division first).

Decompose the integrand into partial fractions to calculate the integral of this rational function with distinct real roots in the denominator:

Determine the integral of the following rational function by previously decomposing it into a sum of simpler fractions:

$\displaystyle \int \dfrac{x + 7}{x^2 + 4x + 3} \, dx$

Calculate the indefinite integral of the following rational function with three distinct real roots in the denominator:

$\displaystyle \int \dfrac{2x^2 - x}{(x-1)(x+1)(x+2)} \, dx$

Calculate the indefinite integral of the following rational function by decomposing it into partial fractions, knowing that its denominator has a repeated real root:

$\displaystyle \int \dfrac{5x - 2}{x^2 - 4x + 4} \, dx$

Determine the values of the parameters $A$, $B$ and $C$ so that the following equality is true and, subsequently, calculate the antiderivative of $f(x)$ that vanishes at $x=0$:

Calculate the antiderivative of the function $f(x) = \dfrac{x^4 + x^3 - 2}{x^3 - x^2}$ that passes through the point $P(2, 5)$. Note that, due to the degrees of the polynomials, you will need to perform polynomial division first and then decompose the remaining fraction considering the existence of repeated real roots.

Determine the indefinite integral of the following function, whose form corresponds to an arctangent-type primitive:

In arctangent-type integrals, it is sometimes necessary to manipulate the denominator. Calculate the following primitive by completing the square in the denominator so that it takes the form of type $\int \frac{u'}{1+u^2} dx$:

$\displaystyle \int \dfrac{3}{x^2 + 2x + 5} \, dx$

Find the family of functions $F(x)$ resulting from integrating the function:

$f(x) = \dfrac{2x + 5}{x^2 + 9}$

Hint: Separate the integral into two parts, one resulting in a natural logarithm and another in an arctangent function.

A team of engineers is monitoring the coolant flow in a thermal power station. The rate of change of the liquid volume with respect to time $t$ is given by the function $v(t) = \dfrac{4t + 10}{t^2 + 4t + 13}$. Find the expression for the total volume $V(t)$ given that at the initial time ($t=0$) the tank contained $100$ gallons.

Note: Observe that the denominator has no real roots, so the solution will combine a natural logarithm and an arctangent.

Calculate the indefinite integral $\displaystyle \int \frac{3x^2 + 1}{x^3 + x^2 + x + 1} \, dx$.

To do this, observe that the denominator can be factored using Ruffini's rule or by grouping terms, resulting in a product of a linear factor and an irreducible quadratic factor.

Determine the indefinite integral of the function $g(x) = \dfrac{1}{x^4 - 1}$. To solve this exercise, you must factor completely over the real numbers and decompose the fraction into a sum that includes terms with distinct real roots and a term with complex roots of the arctangent type.

Find the set of all antiderivatives of the following function, first making a suitable substitution that transforms it into a rational function:

Calculate the antiderivative of the function $h(x) = \dfrac{x^5 + x^2}{x^3 - 1}$. Before integrating, simplify the expression by polynomial division and check if there is any standard identity or factorization that allows you to reduce the complexity of the integrand before applying partial fraction decomposition.

Use the method of integration by parts to solve the following integral. Remember to choose $u$ and $dv$ appropriately following the LIATE mnemonic:

$\displaystyle \int x \cdot e^x dx$

Solve the following integral using the method of integration by parts:

$\displaystyle \int (x + 2) \cdot \cos(x) dx$

Calculate the primitive of the function $f(x) = (2x + 3) \cdot e^x$ whose graph intersects the y-axis ($OY$-axis) at the point $(0, 5)$. Use the method of integration by parts and check the result by differentiating the obtained function.

Find the primitive of the function $g(x) = x^2 \cdot \cos(x)$ that passes through the origin. You will need to apply the method of integration by parts twice consecutively.

Calculate the primitive of the function $g(x) = x^2 \cdot \ln^2(x)$ given that its graph passes through the point $P(1, \frac{2}{27})$. To do this, you will need to apply the method of integration by parts successively.

Find the indefinite integral $\int e^{2x} \cdot \sin(x) \, dx$.

These types of integrals are called "cyclic". Apply the method of integration by parts until the original integral appears again and solve for its value.

In the study of alternating current electrical circuits, the current intensity $i(t)$ in amperes is usually modeled by functions combining exponentials and trigonometrics. Suppose that the rate of change of the charge of a capacitor is given by the function:

Calculate the set of all primitive functions $F(t)$ associated with this intensity using the method of integration by parts. Note that these types of integrals are solved by the "iteration" or "cyclic" method.

Determine a function $f(x)$ whose derivative is $f'(x) = \cos(\ln x)$.

Help: Use the method of integration by parts considering $u = \cos(\ln x)$ and $dv = dx$. After applying it twice, you will obtain an expression where the original integral appears, allowing you to solve for it.

Calculate the primitive of the natural logarithm function using the method of integration by parts, taking $u = \ln(x)$ and $dv = dx$:

$\displaystyle \int \ln(x) dx$

An electronic components company estimates that its marginal cost (the cost of producing one additional unit) follows the function $C'(x) = x \cdot \ln(x)$, where $x$ is the number of thousands of units produced.

Determine the total cost function $C(x)$ given that the fixed costs (cost when $x = 1$, in this simplified model) are $10$ thousand dollars.

Calculate the indefinite integral of the function $h(x) = \text{arctg}(2x)$.

Hint: Use the method of integration by parts taking $u = \text{arctg}(2x)$ and $dv = dx$. Remember that the derivative of $\text{arctg}(f(x))$ is $\frac{f'(x)}{1 + [f(x)]^2}$.

Calculate the primitive of the function $h(x) = \operatorname{arc sen}(\sqrt{x})$.

Hint: First make a substitution of the type $x = t^2$ to eliminate the root and, subsequently, apply the method of integration by parts.

There are integrals that require great technical skill to decompose the integrand before applying the method of parts. Calculate the following indefinite integral using the strategic substitution $u = x \cdot e^x$ at some point in the process or by operating directly by parts:

Find the function $f(x)$ given that its derivative is $f'(x) = \frac{1}{x + 2}$ and that its graph passes through the point $P(0, 5)$.

A team of engineers is analyzing the discharge rate of the battery of a new drone prototype. They have determined that the function representing the rate of change of the charge (in mAh per minute) is given by $f(t) = 3t^2 - \frac{4}{t+1} + \sqrt{t}$, where $t$ is the time in minutes.

Find the total charge function $F(t)$, given that at the initial time ($t = 0$) the drone's charge was $2500$ mAh.

Find the function $f(x)$ that satisfies the condition of being continuous over the entire domain of definition of its branches, given that its derivative is given by:

and that $f(1) = 1$. You will need to integrate the second branch by parts and determine the value of the constants of integration to guarantee continuity at $x = 0$.

A group of young engineers is monitoring the data flow of a server. The growth rate of stored data (in GB/min) is given by the function $v(t) = \frac{4}{t + 1}$, where $t$ is the time in minutes. Obtain the general expression for the data storage function $S(t)$.

An object moves along a straight line with a time-dependent acceleration defined by $a(t) = \ln(t+1)$. If at the initial time ($t=0$) the object was at rest and at the position $s=10$ feet, determine the equation of the position $s(t)$ at any time $t$.

Note: Remember that velocity is the integral of acceleration and position is the integral of velocity. Both integrals require the use of the method of integration by parts.

It is known that the acceleration of a particle moving in a straight line is defined by the expression $a(t) = t \cdot \sin(t)$ for $t \geq 0$.

Determine the velocity function $v(t)$, given that at time $t = \pi$ the velocity of the particle is $2$ ft/s.