Integration of trigonometric functions
$\sin^2 x$, $\tan^3 x\cdot\sec^2 x$ or $\cos(4x-1)$ are not in the table of standard integrals as they stand. But with a trigonometric identity, or by spotting the hidden chain, they turn into something that is. There is no new technique here: it is just the table of standard integrals plus identities from 1st Bachillerato.
First check whether it is the chain in disguise
Find the indefinite integral of the following function, noting that it is a composite function containing the derivative of the argument:…
Find the indefinite integral of the following function, noting that it is a composite function containing the derivative of the argument:
$\int x \cdot \sin(x^2 + 3) \, dx$
Disguised power: $\sin^n x\cdot\cos x$ and similar
Find the indefinite integral of the function $f(x) = \tan^3(x) \cdot \sec^2(x)$. Remember to check whether the expression fits a power-type integral where a function is…
Find the indefinite integral of the function $f(x) = \tan^3(x) \cdot \sec^2(x)$. Remember to check whether the expression fits a power-type integral where a function is raised to an exponent and multiplied by its derivative.
Identities to remove even powers: $\sin^2$, $\cos^2$, $\tan^2$
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…
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…
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.
Combining identity and chain in a single integral
To start practising finding antiderivatives, find the general antiderivative of the following function, which includes a linear adjustment in the argument of the cosine:…
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: $$f(x) =…
Calculate the antiderivative of the following trigonometric function using fundamental identities to simplify the expression before integrating:
$$f(x) = \frac{\cos^3(x)}{1 - \sin(x)}$$