> For the complete documentation index, see [llms.txt](https://jamesbrayy.gitbook.io/atar/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jamesbrayy.gitbook.io/atar/specialist/calculus.md).

# calculus

***

## **integration:**

#### *u substitution*

* $$\int f^n(x) , dx ;;\longrightarrow;; u = f(x), \quad du = a , dx$$
* $$\int a \cos^p(bx) , \sin^q(bx) , dx ;;\longrightarrow;;u = \cos(bx), \quad du = -b \sin(bx) , dx$$

#### *trig substitution*

* ![](/files/WZHV6KKVYDx1RVzIzDaR)

#### *partial fractions*

* given a function $$\frac{f\left(x\right)}{g\left(x\right)}$$ where the degree of $$f$$ is greater than the degree of $$q$$:
  * for distinct linear factors $$g\left(x\right)=\left(x-a\right)\left(x-b\right)\dots$$
    * $$\begin{aligned}\frac{f(x)}{g(x)} &= \frac{A}{x-a} + \frac{B}{x-b} + \dots \f(x) &= A(x-b) + B(x-a) + \dots\end{aligned}$$
  * for repeated linear factors $$g\left(x\right)={\left(x-a\right)}^{n}$$
    * $$\begin{aligned}\frac{f(x)}{g(x)} &= \frac{A}{x-a} + \frac{B}{(x-a)^2} + \dots + \frac{N}{(x-a)^n} \f(x) &= A(x-a)^{n-1} + B(x-a)^{n-2} + \dots + Y(x-a) + Z\end{aligned}$$

#### *volumes of revolution*

* $$\begin{aligned}V &= \pi \int\_a^b f^2(x) , dx \quad \mathrm{or} \quad \pi \int\_c^d f^2(y) , dy \end{aligned}$$
* ![](/files/BBkwACpEzb4AGp30AmQ5)

## **differentiation**

#### *implicit differentiation*

* $$\frac{d}{dx} \big(x^p + y^q \big) = p x^{p-1} + \frac{dy}{dx} , q y^{q-1}$$

#### *logarithmic differentiation*

* ![](/files/A6Mmdqb3dDvaEzD0P94Z)

#### *differential equations*

* just integrate with separation of variables

#### *related rates*

* just write down all parameters at start of question and remember $$\frac{dy}{dx}=\frac{dy}{dz}\times \frac{dz}{dx}$$

#### logistic curves

* $$\begin{aligned}\frac{dP}{dt} &= r P (k - P) \P &= \frac{k}{1 + \left( \frac{k}{P\_0} - 1 \right) e^{-r k t}}\end{aligned}$$
* ![](/files/ymUfOSkQlqDlbElvMhtr)
* to go from $$\frac{dP}{dt}$$ to $$P$$, separate and use partial fractions
* to go from $$P$$ to $$\frac{dP}{dt}$$, use a $$u$$- substitution for the denominator and implicitly differentiate
  * ![](/files/Xfdm1JdwPB9H7NfE2nSx)

## **motion**

#### *rectilinear motion*

* $$a = \frac{dv}{dt} = v \frac{dv}{dx} = \frac{d}{dx}\left(\frac{v^2}{2}\right)$$

#### *simple harmonic motion*

* $$\frac{d^2 x}{dt^2} = -k^2 x$$
  * $$x = A \sin(kt + \phi) \quad \text{or} \quad x = A \cos(kt + \phi)$$
* $$v^2 = k^2 (A^2 - x^2)$$

## **vector calculus**

* $$\begin{aligned} \text{distance travelled from }t\_1 \text{ to } t\_2 &= \int\_{t\_1}^{t\_2} | \underset{\sim}{{v}}(t) | : dt \\&= \int\_{t\_1}^{t\_2} \sqrt{ \big(v\_x(t)\big)^2 + \big(v\_y(t)\big)^2 } , dt\end{aligned}$$
* $$\text{change in displacement} = \int\_{t\_1}^{t\_2} \underset{\sim}{{v}}(t) , dt$$