# Numbers | Maths with Lemon

Source: https://mathswithlemon.io/aa-calculus-further%20integration.html


## Further Integration

"In my free time I do differential and integral calculus. " Karl Marx


#### Integration by substitution


##### What you have to know:

- The integrals of all basic functions (If not, check this lesson)
- 1. Watch the video:
- 2. Watch integrals by substitution with logs video .
- 3. Watch integrals by substitution with polynomial functions video .

#### Integration by parts


##### What you have to know:

- The derivatives of all known functions (If not, check this lesson)
- The integrals of all basic functions (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Watch this special case of definite integrals .

#### Extra


##### Material and references:

- Hodder Book HL(ISBN: 9781510462366) : 10G, 10H

##### Key Points

- 1. You should be able to use the results given in your formula booklet to differentiate and integrate various functions. Standard integrals can be combined with a linear substitution: if \( \int f(x)\,dx = F(x) \), then \[ \int f(ax + b)\,dx = \frac{1}{a}F(ax + b). \] In particular, \[ \int \frac{1}{\sqrt{a^2 - (x + b)^2}}\,dx = \arcsin\!\left(\frac{x + b}{a}\right) + c, \] \[ \int \frac{1}{a^2 + (x + b)^2}\,dx = \frac{1}{a}\arctan\!\left(\frac{x + b}{a}\right) + c. \]
- 2. You should be able to use integration by substitution: differentiate the substitution to express \( dx \) in terms of \( du \); replace all occurrences of \( x \) by the relevant expression in terms of \( u \); change the limits from \( x \) to \( u \); and simplify as much as possible before integrating.
- 3. You should be able to use integration by parts: \[ \int u\,\frac{dv}{dx}\,dx = uv - \int v\,\frac{du}{dx}\,dx \]

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