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:

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

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, $F(0) = 0$.

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 $\sec^2(x)$.