Integration of Rational Fractions and Integration by Parts

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

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

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

Find the antiderivative of the function $g(x)$ by first adjusting the numerator to obtain an integer part and a fractional part:

A market study analyses 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 finding the antiderivative of this function. (Note: Perform polynomial division first).

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 must first perform polynomial division and then decompose the remaining fraction taking into account the presence of repeated real roots.

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

Determine the integral of the following rational function by first 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 by decomposing it into partial fractions, given that its denominator has a double real root:

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

We want to find a function $F(x)$ whose derivative is $f(x) = \dfrac{x^2 - 3x + 2}{(x+1)^3}$. Find this function knowing that its graph passes through the point $(0, 2)$. This problem requires the use of partial fractions with real roots of multiplicity three.

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

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

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

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

To do this, note that the denominator can be factorised using Ruffini's rule or by grouping terms, resulting in the 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 fully factorise the denominator over the real numbers and decompose the fraction into a sum containing terms with simple real roots and a term with complex roots yielding an arctangent form.

A team of engineers is monitoring the flow of coolant in a thermal power plant. The rate of change of the volume of liquid 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 $V(t)$ knowing that at the initial instant ($t=0$) the tank contained $100$ liters.

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

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

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

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

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

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

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

Solve the following integral using the integration by parts method:

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

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

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

This type of integral is 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 modelled using functions combining exponential and trigonometric functions. Suppose the rate of change of the charge on a capacitor is given by the function:

Find the set of all antiderivatives $F(t)$ associated with this current using the method of integration by parts. Keep in mind that this type of integral is solved using the "iteration" or "cyclic" method.

A team of engineers is analysing the discharge rate of a new drone prototype's battery. 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 $F(t)$, knowing that at the initial moment ($t = 0$) the drone's charge was $2500$ mAh.

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

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

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

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

Find the analytical expression of a function $f(x)$ such that its second derivative is $f''(x) = x \times \operatorname{sen}(x)$, knowing that the tangent line to the curve at the origin is the x-axis itself.

It is known that the gradient of the tangent line to a curve at any point $(x, y)$ is given by the expression $m = \dfrac{x^3 + x^2}{x^2 + 1}$. Find the equation of this curve, knowing that it passes through the point $(0, 1)$.