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Integral Generalizations

Just keeping a note to self that I can revise and study for my upcoming integration bee. I'd love your advice for studying for it here!


\(\displaystyle \int x^ne^x \ dx = n!e^x \left( \frac{x^n}{n!} - \frac{x^{n - 1}}{(n - 1)!} + \frac{x^{n - 2}}{(n - 2)!} - \frac{x^{n - 3}}{(n - 3)!} + \cdots + \right) + C\)

\(\displaystyle \begin{align} \int \frac{1}{(x + a)(x + b)} \ dx = \frac{1}{a - b} \ln \left| \frac{x + b}{x + a} \right| + C, & & a > b \end{align}\)

\(\displaystyle I_a = \int \frac{1}{x^a + x} \ dx = -\frac{1}{a - 1}\ln \left(1 + \frac{1}{x^{a-1}}\right) + C\)

\(\displaystyle \begin{align} I_n = \int_0^{\frac{\pi}{2}} \sin ^n (x) \ dx = \int_0^{\frac{\pi}{2}} \cos ^n (x) \ dx, & & n \geq 3 \end{align}\)

  • \(\displaystyle I_{2k} = \frac 12 \times \frac 34 \times \frac 56 \times \cdots \times \frac{2k - 1}{2k} \times \frac{\pi}{2} \ \ \Longleftarrow \ \ \frac 12 \times \cdots \times \frac{\small \text{one less}}{\small \text{the power itself}} \times \frac{\pi}{2}\)
  • \(\displaystyle I_{2k + 1} = \frac 23 \times \frac 45 \times \frac 67 \times \cdots \times \frac{2k}{2k + 1} \ \ \ \ \ \ \ \Longleftarrow \ \ \frac 23 \times \cdots \times \frac{\small \text{one less}}{\small \text{the power itself}}\)

\(\displaystyle I_a = \int_0^1 \frac{x^a - 1}{\ln x} \ dx = \ln (a + 1) + C\)

\(\displaystyle \int e^x \bigg( f(x) + f'(x) \bigg) \ dx = e^xf(x) + C\)

\(\displaystyle \int_0^1 x^m(1 - x)^n \ dx = \frac{m!n!}{(m + n + 1)!} + C\)

Note by Zach Abueg
1 month ago

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