Exercises: Integration of trigonometric functions

To start practising finding antiderivatives, find the general antiderivative of the following function, which includes a linear adjustment in the argument of the cosine:

$f(x) = \cos(4x - 1)$

Find the indefinite integral of the following function, noting that it is a composite function containing the derivative of the argument:

$\int x \times \sin(x^2 + 3) \, dx$

On many occasions, trigonometric identities allow us to transform a difficult expression into an immediate one. Use the relation $\tan^2(x) + 1 = \sec^2(x)$ to calculate the following antiderivative:

$\int \tan^2(x) \, dx$

Determine the family of antiderivatives of the given expression. Keep in mind that the derivative of sine is cosine:

$f(x) = \sin^4(x) \times \cos(x)$

Find the antiderivative of the function $f(x) = \sec^2(3x)$ given that its graph passes through the point $P(0, 2)$. Remember to calculate the constant of integration $C$.

A simple harmonic oscillator moves such that its velocity as a function of time $t$ (in seconds) is given by $v(t) = 3 \times \cos(t)$. If at the initial time $t=0$ the object is at position $s=0$, obtain the expression for its position $s(t)$ by finding the corresponding antiderivative.

Find the following indefinite integral using the natural logarithm in the result, recognising that the numerator is (almost) the derivative of the denominator:

$\int \frac{\sin(x)}{5 + \cos(x)} \, dx$

Obtain the set of all antiderivatives of the following composite function, which combines a square root with a trigonometric function:

$f(x) = \frac{\cos(x)}{\sqrt{\sin(x)}}$

Aoife is calculating the antiderivative of the function $f(x) = \sin^2(x)$ and writes on her exam:

arguing that she has applied the power rule $\int u^n\, du = \frac{u^{n+1}}{n+1} + C$ by taking $u = \sin(x)$ and $n = 2$.

1. a) Explain precisely the conceptual error made by Aoife. What essential factor is missing from the integrand for her deduction from the chain rule to be mathematically valid?
2. b) Correct the exercise by finding the correct family of antiderivatives using the double-angle trigonometric identity.

A continuous function $f(x)$ is known to have a general antiderivative of the form:

where $a$ is a non-zero real constant.

1. a) Knowing that at the point $x = \frac{\pi}{4}$ the original function takes the value $f\left(\frac{\pi}{4}\right) = 8$, deduce the exact value of the parameter $a$ by applying the chain rule to $F(x)$.
2. b) Once the value of $a$ is found, write the algebraic expression of $f(x)$ expressed solely in terms of powers of $\tan(x)$ and $\cos(x)$.

Calculate the antiderivative of the following trigonometric function using fundamental identities to simplify the expression before integrating:

Find the indefinite integral of the function $f(x) = \tan^3(x) \times \sec^2(x)$. Remember to check whether the expression fits a power-type integral where a function is raised to an exponent and multiplied by its derivative.

An architecture firm is designing the profile of a ramp for an extreme sports park. The gradient of the ramp at each point $x$ is given by the function $g(x) = \sin^2(x)$. If the ramp starts at the point $(0, 0)$, determine the function $G(x)$ that describes the height of the ramp at any point.

Find the antiderivative of the function $f(x) = \frac{\sin^3(x)}{\cos^2(x)}$.

Hint: As this is an odd function in $\sin(x)$, use the substitution $t = \cos(x)$.

Determine the indefinite integral of the function $f(x) = \frac{1}{\sin^2(x) \times \cos^2(x)}$.

Hint: You can try substituting the number $1$ in the numerator with the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ to split the integral into two simpler terms.

Find the antiderivative of the function $h(x) = x^2 \times \cos(x)$ using the method of integration by parts. You will need to apply the method twice to reduce the degree of the polynomial until an immediate integral is obtained.

In a physics experiment, the damped oscillation of a spring is analysed, where its vibration velocity as a function of time $t$ follows the function $v(t) = e^{-t} \times \sin(2t)$. Determine the position function $s(t)$, knowing that this is a cyclic integral.

Find the antiderivative $F(x)$ of the function $f(x) = \frac{\sin(x)}{\cos^3(x)}$ that passes through the origin, that is, $F(0) = 0$.

To solve the indefinite integral $\int \sin(x) \times \cos(x)\, dx$, three Leaving Certificate students propose different approaches:

• Aoife makes the direct substitution taking $u = \sin(x)$ and obtains $F_1(x) = \frac{1}{2}\sin^2(x) + C_1$.

• Seán takes $u = \cos(x)$ and obtains $F_2(x) = -\frac{1}{2}\cos^2(x) + C_2$.

• Cian first applies the double-angle identity $\sin(2x) = 2\sin(x)\cos(x)$ and obtains $F_3(x) = -\frac{1}{4}\cos(2x) + C_3$.

1. a) Algebraically prove that all three answers are correct and that they describe exactly the same family of antiderivative functions, determining the exact numerical relationship between the constants $C_1$, $C_2$, and $C_3$.
2. b) If you now had to solve $\int \sin^5(x) \times \cos(x)\, dx$, compare the three previous methods and justify which one of them is the only one that turns out to be immediate and generalizable without resorting to cumbersome calculations. Obtain the final result using that method.

During a summer day at a solar thermal power plant in County Wexford, the instantaneous electrical power fed into the grid (in megawatts, MW) between 10:00 ($t = 0$) and 16:00 ($t = 6$ hours) is modeled according to the function:

The accumulated energy $E(t)$ (in MW·h) since 10:00 is an antiderivative of $P(t)$ that satisfies the initial condition $E(0) = 0$.

1. a) Use the double-angle formula to transform the expression of $P(t)$ and integrate step by step to obtain the function $E(t)$, paying attention to the constant introduced by the inner derivative of the linear argument.
2. b) Find the total accumulated energy at the end of the period, $E(6)$, leaving the result in terms of $\pi$ and calculating its approximate value to two decimal places. From this datum, determine and interpret the average power supplied by the plant during that 6-hour interval.

In the design of alternating current circuits, instantaneous power is usually given by the product of exponential and trigonometric functions. Calculate the family of antiderivatives of the following expression, which represents the rate of change of energy in a specific component:

Tip: First use a trigonometric identity to reduce the exponent of the cosine before applying the integration by parts method.

Calculate the indefinite integral of the following trigonometric rational function. This type of integral frequently appears in the calculation of gravitational fields in astrophysics:

Note: Consider a substitution relating the trigonometric functions present and remember the fundamental trigonometric identity.

Determine the antiderivative of the function $f(x)$ knowing that its graph passes through the point $P\left(\frac{\pi}{4}, 1\right)$ and its derivative is:

$f'(x) = \frac{1}{\sin^4(x) \times \cos^2(x)}$

Hint: Try to express the numerator using the fundamental identity $\sin^2(x) + \cos^2(x) = 1$ iteratively or look for a substitution based on the tangent.

The "universal substitution" is a powerful tool for solving integrals of rational functions of sine and cosine. Use the substitution $t = \tan\left(\frac{x}{2}\right)$ to find the following antiderivative:

Find the antiderivative of the function $f(x)$ described below, using product-to-sum transformations:

Once the family of antiderivatives is obtained, determine the one whose graph has an inflection point with a horizontal tangent at the point with abscissa $x = \frac{\pi}{2}$ and ordinate value $y = 2$.

In the study of the oscillations of a suspension bridge, the acceleration of a point on the structure is given by the function $a(t)$. Find the velocity function $v(t)$ knowing that the initial velocity is $v(0) = 0$ and the acceleration is:

$a(t) = \frac{\sec^2(t)}{\sqrt{4 - \tan^2(t)}}$

Hint: Be careful with the domains of the resulting functions.

Calculate the indefinite integral of the function $f(x)$, which combines powers of inverse trigonometric functions and direct functions:

Find the family of antiderivatives of the following function using an appropriate substitution for even trigonometric functions in sine and cosine:

Suggestion: Divide the numerator and denominator by $\cos^2(x)$ to express the function in terms of $\tan(x)$ and its derivative $\sec^2(x)$.

Consider the function $f(x) = \frac{\cos^3(x)}{\sqrt{\sin(x)}}$, defined on the open interval $\left(0, \frac{\pi}{2}\right)$.

1. a) Before calculating any antiderivative, make a reasoned estimate of the behavior of any antiderivative $F(x) = \int f(x)\, dx$ as $x \to 0^+$. To do so, take into account that for values very close to zero the approximations $\cos(x) \approx 1$ and $\sin(x) \approx x$ hold; based on this, deduce whether $F(x)$ is expected to tend to a finite value or diverge to infinity when approaching the origin.
2. b) Use the fundamental identity $\cos^2(x) = 1 - \sin^2(x)$ to decompose the integral into a sum of immediate integrals of power type $u^n \times u'$, and calculate the algebraic expression of the general family of antiderivatives $F(x)$.
3. c) Calculate the limit $\lim_{x \to 0^+} F(x)$ from the obtained expression and check whether your initial qualitative estimate was correct, reasoning the consistency between the local analysis of the integrand and the result of the integration.

The goal is to determine a differentiable function $f(x)$ on the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ whose graph passes through the origin, $f(0) = 0$, and whose derivative function corresponds to the expression:

1. a) A student tries to integrate each term separately searching for formulas for fourth powers, encountering serious difficulties. Prove that, by applying the factorization of a difference of squares and the fundamental identity $\sec^2(x) - \tan^2(x) = 1$, the derivative immediately reduces to $f'(x) = 2\sec^2(x) - 1$. Justify why transforming the integrand before operating radically simplifies the problem.
2. b) Integrate the simplified expression and find the explicit formula of the function $f(x)$ that satisfies the condition $f(0) = 0$.
3. c) Without calculating the second derivative $f''(x)$, deduce the intervals of concavity and convexity of $f(x)$ on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, as well as the existence of inflection points, by exclusively analyzing the increase, decrease, and relative extrema of $f'(x)$ from the properties of the cosine function.

Topic Challenge. How would you model the daily energy flow in our country?

1. Data collection: Go to the EirGrid Smart Grid Dashboard website and download the hourly all-island electricity demand data for a recent day.

2. Modelling: In GeoGebra or a spreadsheet, plot the time series $t \in [0, 24]$. Design a continuous model $f(t) = k \left( \sin\left(\frac{\pi t}{12}\right) + \frac{1}{t^2 + 4} + 1.1 \right)$ and find the constant $k$ so that it satisfies the conditions of a probability density function of a continuous random variable.

3. Analytical integration: Obtain the distribution function $F(t) = \int_{0}^{t} f(u)\,du$ applying elementary decomposition, trigonometric antiderivatives, and rational function antiderivatives. From $F(t)$, calculate the probability that daily consumption is concentrated during peak hours, between $19.0$ and $22.0$ hours.

4. Final report: Write a short report with the graphs of $f(t)$ and $F(t)$, the algebraic development of the integrals, and a reasoned reflection on the fit to real data.

The following will be assessed:

• Analytical mastery in computing immediate, rational, and trigonometric integrals.

• Rigour in determining parameters and validating the properties of the continuous random variable ($f$ and $F$).

• Technical skill in GeoGebra or spreadsheet software and critical contrast against the real EirGrid series.