Exercises: Indefinite integral. Elementary decomposition and standard integrals

A professional skater is analyzing the trajectory of his latest jump on a ramp. The function that describes the variation of his height with respect to time (vertical velocity) is given by the expression $v(t) = 3t^2 - 4t + 5$, whose graph appears in the figure. Find the family of functions $H(t)$ that represent the skater's height, by calculating the indefinite integral:

In a study on how news spreads on a social network, it was determined that the growth rate of "shares" follows a power law. To find the cumulative total, a programmer must solve the following integral of an irrational function. Calculate it by first expressing the root as a fractional exponent:

Determine whether the function $F(x) = x \times e^x$ is an antiderivative of the function $f(x) = (x+1)e^x$. To justify your answer, remember to use the fundamental relationship between a function and its antiderivative via differentiation.

A hearing study for teenagers analyses the intensity of a sound signal. The function describing the instantaneous sound pressure is $p(x) = \frac{3}{x} + e^{2x}$. Find the general antiderivative of this function by carrying out the integration:

Calculate the indefinite integral of the following combination of trigonometric functions and constants, applying linearity properties (factoring out constants and splitting sums):

Find all antiderivatives of the function $f(x) = \frac{x^4 - 2x^2 + 5}{x^2}$.

In the design of a smartphone camera lens, the curvature function needs to be known. The derivative of this function is $g'(x) = \frac{7}{1 + x^2} - 2^x$. Obtain the expression for the function $g(x)$ by evaluating its indefinite integral:

Find the antiderivative of the function $f(x) = 2x - 6$ whose graph passes through the point $P(3, 4)$.

Note: First calculate the indefinite integral to obtain the family of functions with the constant $C$, and then use the point $P$ to find the specific value of that constant.

Find the family of antiderivatives of the following function, simplifying the result by eliminating negative exponents and using radicals:

$f(x) = \dfrac{3}{x^4} - \dfrac{2}{\sqrt[3]{x^2}} + \dfrac{1}{5x}$

Find the indefinite integral of the function $g(x)$, making the necessary constant adjustments to obtain a logarithmic form:

$I = \displaystyle\int \dfrac{3x - 1}{x^2 - \frac{2}{3}x + 5} \, dx$

A surveillance drone moves such that its horizontal velocity (in m/s) is given by the function $v(t) = 4e^{2t-1} + \sin(3t)$. Given that at the initial instant $t=0$ the drone is at position $s=0$, determine the expression for the position function $s(t)$.

Find the antiderivative of the function $f(x) = \dfrac{x^2 - 4x + 1}{x-2}$ whose graph passes through the point $P(3, 5)$.

Note: Perform polynomial division first.

Determine the indefinite integral of the following composite function, identifying the derivative of the inner function:

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

Find the antiderivative $F(x)$ of the function $f(x) = \dfrac{1}{\cos^2(3x)} + 5^{2x}$ knowing that $F(0) = 1$.

Calculate the indefinite integral of the following function using linearity decomposition and adjusting for the derivative of the exponent:

$I = \displaystyle\int \left( 2x \times e^{x^2+3} - \dfrac{5}{1+4x^2} \right) \, dx$

Given the function $f(x) = \dfrac{\sin(x)}{\sqrt{1 + \cos(x)}}$, calculate its general antiderivative. Check your result by differentiating the obtained function.

A sound engineer is analysing the signal of a new synthesiser. The rate of change of the wave intensity with respect to time, $t$, is given by the function $f'(t) = \dfrac{e^{2t} + 3e^t}{e^t + 1}$.

Knowing that for a time $t = 0$ the intensity of the wave is $10$ units, determine the algebraic expression for the intensity $f(t)$ by performing the necessary algebraic manipulations to solve the integral.

Find the family of antiderivatives of the following trigonometric function using fundamental identities to transform the integrand into a sum of standard integrals:

Determine the function $f(x)$ whose second derivative is $f''(x) = \dfrac{12x^2 - 4}{(x^2 + 1)^3}$.

It is known that the graph of $f(x)$ has an inflection point with positive $x$-coordinate at which the tangent line is horizontal, and that the graph of $f(x)$ passes through the point $(0, 5)$.

Note: This exercise requires an expert handling of the chain rule in reverse for rational functions.

A logistics company studies the tire wear of its fleet. The wear rate $V(x)$ (in mm per 1000 km) depends on the load $x$ (in metric tons) according to the function:

$V(x) = \frac{x + 1}{\sqrt{4 - (x - 3)^2}}$

Find the accumulated wear function $D(x)$, knowing that for a load of $3$ metric tons ($x = 3$) the wear is zero.

Find the indefinite integral of the following rational function by completing the square in the denominator to identify an arctangent form:

Given the function $f(x) = |x - 2| \times e^x$, find the antiderivative $F(x)$ satisfying $F(0) = 1$.

Remember that for $F(x)$ to be an antiderivative on its entire domain, it must be a continuous function, so you will need to adjust the constants of integration across the different intervals.

Determine the values of the parameters $A, B$, and $C$ such that the function $F(x) = (Ax^2 + B) \sin(2x) + Cx \cos(2x)$ is an antiderivative of $f(x) = x^2 \cos(2x)$.

Set up the system of equations by differentiating $F(x)$ and equating coefficients.

Calculate the indefinite integral of the function:

This exercise assesses the ability to recognize the structure $g(u(x)) \times u'(x)$ when there are multiple composite functions.