Exercises: Integration of trigonometric functions

To start practicing calculating antiderivatives, find the general primitive of the following function which includes a linear adjustment in the argument of the cosine:

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

Calculate the indefinite integral of the following function, noting that it is a composite function where the derivative of the argument appears:

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

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

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

Determine the family of primitive functions of the given expression. Bear in mind that the derivative of sine is cosine:

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

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

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

Calculate the following indefinite integral using the natural logarithm as the result, recognizing that the numerator is (almost) the derivative of the denominator:

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

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

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

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

Find the indefinite integral of the function $f(x) = \tan^3(x) \cdot \sec^2(x)$. Remember to observe if 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.

Hint: Use the double angle formula to reduce the degree of the function.

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

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

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

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

Find the primitive of the function $h(x) = x^2 \cdot \cos(x)$ using the method of integration by parts. You will need to apply the method twice to reduce the degree of the polynomial until you obtain a standard integral.

In a physics experiment, the damped oscillation of a spring is analyzed, whose vibration velocity as a function of time $t$ behaves according to the function $v(t) = e^{-t} \cdot \sin(2t)$. Determine the position function $s(t)$ given that this integral is cyclic.

Calculate the primitive $F(x)$ of the function $f(x) = \frac{\sin(x)}{\cos^3(x)}$ that satisfies the condition of passing through the origin of coordinates, that is, $F(0) = 0$.

Hint: Identify whether it is a logarithmic or power-type integral before applying the initial conditions.

In the design of alternating current circuits, instantaneous power is usually given by the product of exponential and trigonometric functions. Calculate the family of primitive functions 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 method of integration by parts.

Calculate the indefinite integral of the following rational trigonometric function. These types of integrals frequently appear in the calculation of gravitational fields in astrophysics:

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

Determine the primitive of the function $f(x)$ given that its graph passes through the point $P(\pi, 1)$ and its derivative is:

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 primitive:

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

Once the family of primitives is obtained, determine the one whose graph has a relative maximum at the point with x-coordinate $x = \frac{\pi}{2}$ and y-coordinate $y = 2$.

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 position function $s(t)$ given that the initial velocity is $v(0) = 0$, the initial position is $s(0) = 1$ and the acceleration is:

Help: You will need to integrate twice. The first integral will give you the velocity and the second the position. 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:

Hint: Decompose the integral into a sum of two simpler integrals and analyze whether either of them can be solved as an integral of the type "function multiplied by its derivative".

Find the family of primitives of the following function using a suitable substitution for even trigonometric functions in sine and cosine:

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