Maths - Integration by Substitution
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Flashcards
\[\int (x+2)^5 dx\]
What subsitution could you make in terms of $u$?
\[u = x + 2\]
If $u = x + 2$, what is $\frac{du}{dx}$?
\[\frac{du}{dx} = 1\]
If $u = x + 2$ and $\frac{du}{dx} = 1$, what is $dx$?
\[du\]
\[\int (x+2)^5 dx\]
If $u = x+2$ and $du = dx$, how could you rewrite the integral?
\[\int u^5 du\]
\[\int u^5 du\]
What is this equal to?
\[\frac{u^6}{6} + c\]
\[\frac{u^6}{6} + c\]
Can you reverse the subsitution $u = x + 2$?
\[\frac{(x + 2)^6}{6} + c\]
2021-03-30
If the subsitution for an integral is $u = x^2$ and the bounds are $[2, 4]$, what are the new bounds?
\[[4, 16]\]
\[\int \sqrt{1 + x^2}\]
What subsitution would you make?
\[u = \sinh x\]
\[\int \sqrt{1 - x^2}\]
What subsitution would you make?
\[u = \sin x\]
or
\[u = \cos x\]\[\int \sqrt{x^2 - 1}\]
What subsitution would you make?
\[u = \cosh x\]
\[\int \sqrt{4 + 4^2}\]
What subsitution would you make?
\[u = 2\sinh x\]
2022-04-12
\[\int \frac{1}{2} x^3 \sqrt[3]{4 - x^2} \text{d}x\]
What substitution should you make here?
\[u = 4 - x^2\]