Exercises: Indefinite integral. Elementary decomposition and standard integrals

A professional skater is analyzing the trajectory of his last jump on a ramp. The function describing the variation of his height with respect to time (vertical velocity) is given by the expression $v(t) = 3t^2 - 4t + 5$. Find the family of functions $H(t)$ representing the skater's height by calculating the indefinite integral:

In the study of the spread of news on a social network, it has been determined that the growth rate of "shares" follows a power law. To find the accumulated total, a programmer must solve the following integral of an irrational function. Calculate it by first converting the radical to a fractional exponent:

Determine whether the function $F(x) = x \cdot e^x$ 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 through differentiation.

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

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

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

Hint: To solve it immediately, decompose the fraction into three simple terms before integrating.

In the design of a lens for a smartphone camera, it is necessary to know the curvature function. The derivative of this function is $g'(x) = \frac{7}{1 + x^2} - 2^x$. Obtain the expression for the function $g(x)$ by calculating 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 this 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 structure:

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

A surveillance drone moves such that its horizontal speed (in ft/s) is given by the function $v(t) = 4e^{2t-1} + \sin(3t)$. Given that at the initial time $t=0$ 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 $f(x) = \dfrac{1}{\cos^2(3x)} + 5^{2x}$ given that $F(0) = 1$.

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

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

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 analyzing the signal of a new synthesizer. 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 wave intensity is $10$ units, determine the analytical expression for the intensity $f(t)$ by carrying out the necessary algebraic manipulations to solve the integral.

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

Hint: Try to express the numerator in terms of sine using the fundamental trigonometric identity.

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 a point of inflection where the tangent line is horizontal and passes through the point $(0, 5)$.

Note: This exercise requires expert use of the reverse chain rule for rational functions.

A logistics company is studying the wear of its fleet's tires. The rate of wear $V(x)$ (in inches per 1,000 miles) depends on the load $x$ (in tons) according to the function:

Find the accumulated wear function $D(x)$, knowing that for a zero load ($x=0$) the wear is zero.

Hint: Decompose the fraction to obtain an integral of root type (power) and another of arcsine type.

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

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

Remember that for $F(x)$ to be an antiderivative over its entire domain, it must be a continuous function, so you will have to adjust the constants of integration in the different pieces.

Determine the values of the parameters $A, B$ and $C$ so that the function $F(x) = (Ax^2 + Bx + C) \cdot \sin(2x)$ 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 $g(u(x)) \cdot u'(x)$ when there are multiple composite functions.