King’s rule \[ \int_a^b f(x) \,\mathrm{d}x = \int_a^b f(a + b - x) \,\mathrm{d}x \]
Proof \[ \int_a^b f(x) \,\mathrm{d}x = \int_a^b f(a + b - x) \,\mathrm{d}x \]
Common tricks Adding both of the integral By adding the left and right side of the integral, you can see that the original integral …
2026-08-31123 words1 min
integral sin(ln(x))
Statement \[ \int \sin{\left(\ln{x}\right)} \, \mathrm{d}x \]
Solution solution 1: u-sub \begin{align*} u &= \ln{x} \\ e^u &= x \\ \frac{\mathrm{d}}{\mathrm{d}u} e^u &= \mathrm{d}x \\ \end{align*}
Then we u-sub:
\begin{align*} I &= \int \sin{\left(\ln{x}\right)} \, \mathrm{d}x \\ …
2026-08-31314 words2 mins
integral e^x sin(x)
Statement \[ \int e^x \sin{x} \, \mathrm{d}x \]
Solution Solution 1: integrate by parts For the first integral, we let \(u = \sin{x}\) and \(\mathrm{d}v = e^x\) which follows \(\int u \mathrm{d}v = uv - \int v \mathrm{d}u \). Similarly for the second integral.
\begin{align*} I &= \int e^x …
