Calculus 2, Fall 2016

Study tips

  • When learning techniques of integration, always make guesses based on what you already know and test them out! Compare what works with what doesn't work (and explain why).

  • Don't just read the textbook. Instead, read an example in your textbook quickly, scanning it for main ideas. Write down the problem statement, flip the page over so you can't see the solution, and try to write the solution on your own. Flip back only when you're really stuck. This method of active studying (as opposed to passive reading) forces you to recall prior knowledge and construct new knowledge yourself, both of which build strong neural connections.

  • There are no dumb questions – only springboard questions.



Prelude to integration by parts

  1. Does \((fg)' = f'g'\)? Explain.

  2. Is it possible to undo the product rule for derivatives?

Integration by parts

Let \(u\) and \(v\) be differentiable functions of \(x\) on an interval containing \(a\) and \(b\). Then, \[\int u \, dv = uv - \int v \, du\] and \[\int_a^b u \, dv = \bigg\lbrack uv \bigg\rbrack_a^b - \int_a^b v \, du.\]

More hills than Boston Marathon

  1. For the function \(y = x \sin(x)\), find the signed area of the first hill, second hill, and \(n\)th hill.

  2. Find \(\displaystyle \int x^3 \sin(x^2) \, dx\).


Graph \(y = \displaystyle \frac{x}{e^x}\). Then, evaluate these integrals.

  1. \(\displaystyle \int_0^3 \frac{x}{e^x} \, dx\)

  2. \(\displaystyle \int (x+1) e^{2x} \, dx\)

  3. \(\displaystyle \int x^3 e^{x^2} \, dx\)


  1. \(\displaystyle \int t^2 e^{3t} \, dt\)

  2. \(\displaystyle \int e^{3x} \cos(2x) \, dx\)

Use known derivative formulas!

  1. \(\displaystyle \int \sqrt{x} \ln(x) \, dx\)

  2. \(\displaystyle \int \arcsin(x) \, dx\)

Use trig identities!

Calculate \(\displaystyle \int \cos^2(\theta) \, d\theta\) in two ways:

  1. By parts, using \(\cos^2(\theta)+\sin^2(\theta) = 1\).

  2. By substitution, using the power-reduction formula \[\cos^2(\theta) = \frac{ 1+\cos(2\theta)}{2}.\]