Integration of trigonometric functions

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

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

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.

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:

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

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 $t = \cos(x)$.

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.

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

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

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

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

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

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 $\sec^2(x)$.

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 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 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 $y = 2$.

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: