Further Maths - Weierstrass Substitution
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 $\tan(4x)$?
\[\frac{1}{4 + 4t^2}\]