18.01 Section, November 9, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Integration techniques: parts, substitution; separable ODE’s
1. Evaluate the following integrals:
Z 4
2
(a)
x ex dx
2
Z
(b)
x√
t ln t dt
(assume x > 1)
1
Z
(c)
π
sin3 x dx
0
2. Instead of actually doing the following integrals (or even picking up a pen), just talk to
your neighbor about how you would go about doing them – e.g. “integration by parts with
f = sin x and g = 2x”.
Z
Z
Z
Z
x
1
dx
dx
t sin 2t dt
x ln x dx
2
1+x
1 + x2
Z
1 + x2
dx
x
Z
Z
2
x3 ex dx
tan x dx
1
3. Find the general solution for
dy
dx
= y 2 + 1.
4. Make up an integration by substitution problem and give it to your neighbor.
Of course, you should actually know how to do the problem before giving it to someone else!
Z
5. Bonus problem:
ln x dx
Hint: integration by parts.
Z
6. Bonus problem:
√
sin x dx
Hint: first make a substitution.
Review
• Integration by parts:
R
f g dx = F g −
R
F g 0 dx where F =
2
R
f.