Maths - Integration by Substitution
Pearson Edexcel Mathematics 2022
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Flashcards
A worked linear substitution
\[\int (x+2)^5 dx\]
What substitution could you make in terms of $u$?
\[u = x + 2\]
\[\int (x+2)^5 dx\]
If $u = x+2$ and $du = dx$, how could you rewrite the integral?
\[\int u^5 du\]
\[\frac{u^6}{6} + c\]
Can you reverse the substitution $u = x + 2$?
\[\frac{(x + 2)^6}{6} + c\]
Changing limits
If the substitution for an integral is $u = x^2$ and the bounds are $[2, 4]$, what are the new bounds?
\[[4, 16]\]
Trigonometric and hyperbolic substitutions
\[\int \sqrt{1 + x^2}\]
What substitution would you make?
\[u = \sinh x\]
\[\int \sqrt{1 - x^2}\]
What substitution would you make?
\[u = \sin x\]
or
\[u = \cos x\]\[\int \sqrt{x^2 - 1}\]
What substitution would you make?
\[u = \cosh x\]
\[\int \sqrt{4 + 4^2}\]
What substitution would you make?
\[u = 2\sinh x\]
Substitution inside a product
\[\int \frac{1}{2} x^3 \sqrt[3]{4 - x^2} \text{d}x\]
What substitution should you make here?
\[u = 4 - x^2\]