Reminders

• MyOpenMath Homework due tomorrow.
• Written homework due Monday in class.
• Quiz on Monday!

Reverse product rule

\[\int f^\prime(x)g(x) dx + \int f(x) g^\prime(x) dx = f(x) g(x) + C\]

Or:

\[\int f(x) g^\prime(x) dx = f(x) g(x) - \int f^\prime(x) g(x) dx + C\]

Substitute:

• \(u = f(x)\), \(du = f^\prime(x) dx\)
• \(v = g(x)\), \(dv = g^\prime(x) dx\)

Then:

\[\int u dv = uv - \int v du\]

Integration by parts formula!

Example

\(\int x e^x dx\). Must pick \(u\) and \(dv\).

• Let \(u = x\), \(dv = e^x dx\).
• Then \(du = dx\), \(v = e^x\)
• \(uv - \int v du = xe^x - \int e^x dx\)
• Or just \(xe^x - e^x + C\)

Q: What if we chose \(u = e^x\), \(dv = x dx\)?

Hmm…

• \(u = e^x\), \(dv = x dx\)
• \(du = e^x dx\), \(v = \frac{x^2}{2}\)
• \(uv - \int vdu = \frac{x^2}{2} e^x - \int \frac{x^2}{2} e^x dx\)
• More complicated than original problem!
• Pick \(u\) and \(v\) carefully!

More guidance

• For \(u\), prefer:
  • logarithms (\((\ln|x|)^\prime = \frac{1}{x}\))
  • inverse trig functions
  • polynomials / algebraic functions
• For \(dv\):
  • trig (sin / cos)
  • exponential functions

For \(u\): LIPET. Logs > Inverse Trig > Polynomials > Exponentials > Trig.

Exercises

1. \(\int x \ln(x) dx\)
2. \(\int \frac{\ln(x)}{x^2} dx\)

Solution (1)

\[\int x \ln(x) dx\]

• Let \(u = \ln(x)\), \(dv = x dx\)
• \(du = \frac{1}{x} dx\), \(v = \frac{x^2}{2}\)
• \(uv - \int v du = \frac{x^2}{2} \ln(x) - \int \frac{x}{2} dx\)
• Or \(\frac{x^2}{2}\ln(x) - \frac{x^2}{4} + C\)

Solution (2)

\[\int \frac{\ln(x)}{x^2} dx\]

• \(u = \ln(x)\), \(dv = \frac{dx}{x^2}\)
• \(du = \frac{1}{x} dx\), \(v = -\frac{1}{x}\)
• \(-\frac{\ln(x)}{x} - \int -\frac{1}{x^2} dx\), or
• \(-\frac{\ln(x)}{x} + \int \frac{1}{x^2} dx\)
• Final answer: \(-\frac{\ln(x)}{x} - \frac{1}{x} + C\)