# Maths - Integration

> Source: https://ollybritton.com/notes/a-level/maths/topics/integration/ · Updated: 2020-10-06 · Tags: maths, school

## See Also
- [Maths - Differentiation](https://ollybritton.com/notes/a-level/maths/topics/differentiation/)
- [Further Maths - Volumes of Revolutions](https://ollybritton.com/notes/a-level/further-maths/topics/volumes-of-revolutions/)

## Flashcards
##### What is $x^n$ integrated??
$$
\frac{x^{n+1}}{n+1} + c
$$

##### What is $kx^n$ integrated??
$$
\frac{kx^{n+1}}{n+1} + c
$$

##### Integrate $2x+4$??
$$
x^2 + 4x + c
$$

##### What does $c$ mean in integration??
The constant of integration.

##### Why is a constant of integration important??
Because it represents any constant term that would disappear when being differentiated.

##### What does it mean to find a particular solution of an integral??
Calculating the actual value of an integral but solving for the constant of integration.

##### If a general solution specifies a family of curves, what does a particular solution specify??
A single curve.

##### If you have a general solution to an integral and is told that the solution passes through a point, what are you then finding??
A particular solution.

##### If $\int 2x+4 dx = x^2 + 4x + c$ but the actual solution passes through $(1,9)$, how can you solve for $c$??
$$
1^2 + 4 \times 1 + c = 9 \\\\
1 + 4 + c = 9 \\\\
5 + c = 9 \\\\
c = 4
$$

##### What's the flow chart process for finding a particular solution to an integral??
$$
\text{integrate} \to \text{substitute} \to \text{solve}
$$

##### What is the first step for evaluating an integral $\int^b_a$??
Finding the antiderivative/indefinite integral $F(x)$.

##### If $f(x)$ is a function, what is the notation for the antiderivative of $f(x)$??
$$
F(x)
$$

##### When evaluating $\int^b_a$, what do you do with $F(x)$??
$$
F(b) - F(a)
$$

##### How can you visualise $F(b) - F(a)$ when finding the area under a curve??
Finding the area up to the upper bound $b$ and then subtracting the unneccesary area up to $a$.

##### Why does the integral symbol look like $\int$??
It's like a long S shape, representing a *s*um.

### 2021-06-09
##### $$\int \tan x dx$$ What is this equal to??
$$
\ln(\sec x) + c
$$

##### What's another way of writing $\ln(\sec x) + c$??
$$
-\ln(\cos x) + c
$$

##### $$\int \cot x dx$$ What is this equal to??
$$
\ln(\sin x) + c
$$

##### $$\int \sec x dx$$ What is this equal to??
$$
\ln(\sec x + \tan x) + c
$$

##### $$\int \csc x dx$$ What is this equal to??
$$
\ln(\csc x - \cot x) + c
$$

### 2021-07-03
##### What is $\sin^2(x)$ in terms of $\cos(2x)$??
$$
\frac{1}{2} (1 - \cos(2x))
$$

##### What is $\cos^2(x)$ in terms of $\cos(2x)$??
$$
\frac{1}{2} (1 + \cos(2x))
$$

##### How can you integrate $\cot^2 x$??
Rewrite as

$$
\csc^2 x - 1
$$

### 2021-12-15
##### What is the formula for the area between the $x$-axis and a parametric curve defined with $x = f(t)$ and $y = g(t)$??
$$
\pi \int^{t = p}_{t = q} y \frac{\text{d}x}{\text{d}t} dt
$$

##### What is the formula for the area between the $y$-axis and a parametric curve defined with $x = f(t)$ and $y = g(t)$??
$$
\pi \int^{t = p}_{t = q} x \frac{\text{d}y}{\text{d}t} dt
$$

##### If you normally use $\int y \text{d}x$ when finding the area between the $x$-axis and a curve, how would this change for integrating parametrically??
$$
\int y \frac{\text{d}x}{\text{d}t} \text{d}t
$$

##### If you normally use $\int x \text{d}y$ when finding the area between the $y$-axis and a curve, how would this change for integrating parametrically integrating parametrically??
$$
\int x \frac{\text{d}y}{\text{d}t} \text{d}t
$$

### 2022-01-08
##### What is the integrand??
The inside of an integral, what is being integrated.

##### What's the difference between the integral and the integrand??
The integrand is the function being integrated, whereas the integral is the whole expression.

### 2022-01-19
##### When doing a contextual integration question, what is dangerous??
Assuming that the object is around the origin.

##### What must you consider every time you do a contextual integration question where you're modelling an object??
Where it is in relation to the origin.

### 2022-04-12
##### $$\int \sin(4x) (1 - \cos 4x)^3 \text{d}x$$ You could overcomplicate this by using several different trig identities and expanding. What could you also do??
Just notice that the derivative of the $\cos$ part is the four times the $\sin$ part.

### 2022-04-17
##### $$\int \frac{1-t^2}{1+t^2} \text{d} x$$ How should you tackle this??
Algebraic long division.

### 2022-05-11
##### $$\int \frac{3x}{\sqrt{4-x^2}} \text{d}x$$ How do you integrate this??
Consider

$$
\frac{\text{d}}{\text{d}x} \sqrt{4-x^2}
$$

##### Why is $$\int \frac{3x}{\sqrt{4-x^2}} \text{d}x$$ equal to $$3\sqrt{4-x^2}$$ and not $$-\frac{3}{2} \sqrt{4-x^2}$$??
Because the $-\frac{1}{2}$ comes from the power rule and the chain rule.

### 2022-05-30
##### $$\int\frac{x^2}{1 + 16x^2}\text{d}x$$ How would you tackle this??
Algebraic long division.

### 2022-06-06
##### $$\lim_{\delta x \to 0} \sum^6_{x = 2} \frac{1}{x} \delta x$$ Can you write this as an integral??
$$
\int^6_2 \frac{1}{x} \text{d}x
$$

##### $$\int^10_3 3x^2 - 4 \text{d}x$$ Can you rewrite this as the limit of a sum??
$$
\lim_{\delta x \to 0} \sum^{10}_{x \to 3} (3x^2 - 4) \delta x
$$

### 2022-06-07
##### $$\sin 6x \sin 8x$$ Rather than integrating by parts, how could you rewrite this in order to help with integration??
$$
\frac{1}{2}\left( \cos(2x) - \cos(14x) \right)
$$

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