# Further Maths - Weierstrass Substitution

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/weierstrass-substitution/ · Updated: 2021-12-01 · Tags: school, further-maths, year-2, further-pure-1

## Flashcards
### 2021-12-01
##### What is the Weierstrass substitution??
The substitution $t = \tan\frac{x}{2}$ used to evaluate integrals.

##### What do you substitute for $\text{d}x$ in the Weierstrass substitution??
$$
\text{d}x = \frac{2}{1 + t^2} \text{d}t
$$

##### Why is the Weierstrass substitution useful??
Because it turns complicated integration of trig functions into a rational function.

##### What technique would you use for evaluating $$\int^{\pi/2}_{pi/3} \frac{1}{1 + \sin x - \cos x} \text{d}x$$??
The Weierstrass substitution.

##### $$\int \csc (x) \text{d}x$$ Can you make the Weierstrass substitution??
$$
\int \left(\frac{1 + t^2}{2t}\right) \left(\frac{2}{1 + t^2}\right) \text{d}t
$$

##### $$\int \frac{\sec x}{1 + \tan x} \text{d}x$$ Can you make the Weierstrass substitution here??
$$
\int \frac{\left(\frac{1 + t^2}{1 - t^2}\right)}{1 + \left(\frac{2t}{1 - t^2}\right)} \left( \frac{2}{1 + t^2} \right) \text{d}t
$$

### 2022-01-20
##### Why must you be careful doing something like a Weierstrass substitution for like $\tan(3x)$??
Because the $\text{d}x$ changes and isn't just $\frac{2}{1 + t^2}\text{d}t$

##### What would be the $\text{d}x$ for a Weierstrass substitution of $\tan(4x)$??
$$
\frac{1}{4 + 4t^2}
$$

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