Mathematics 134 – Intensive Calculus for Science 2 Discussion 1 – Practice on Integration by Substitution February 8, 2006

Background

Last time, we introduced the technique of integration by substitution. Recall that the idea was that many integrals could be reduced to one of our basic forms by a substitution u = g(x), du = g′(x) dx. If the original integral is

∫ f′(g(x))g′(x) dx

(possibly up to a constant factor), and we do this,

∫ f′(u) du = f(u) + C = f(g(x)) + C

(Note: This will work as long as ∫ f′(x) dx is known from basic derivative rules. For instance: To find

∫ x\sqrt{1 − 3x^{2}} dx,

try u = 1 − 3x^{2}, so du = −6x. Hence the x dx is \frac{−1}{6}du, and

∫ x\sqrt{1 − 3x^{2}} dx = \frac{−1}{6} ∫ \sqrt{u} du

= \frac{−1}{6} \frac{2}{3}u^{3/2} + C

= \frac{−1}{9}(1 − 3x^{2})^{3/2} + C

Today, we want to practice on a number of additional examples of this method. (There’s no other way to master this than to “practice, practice, practice!”)

Problems

For each integral, identify a likely u, compute du, and substitute into the integral to obtain an equivalent integral entirely in terms of u. Integrate, then convert back to x.

A)

∫ \sqrt{sin(4x)} cos(4x) dx

B)

∫ (3x + 3)^{34} dx

∫ \frac{x + 1}{x^{2} + 2x + 29} dx

D)

∫ \frac{e^{\sqrt{x}}}{\sqrt{x}} dx

E) A definite integral – compute and explain why your answer is reasonable: ∫_{0}^{2} sin(5πx) dx

F) You’ll need substitution, plus some algebra for this one:

∫ (x + 2)\sqrt{x − 1} dx

Assignment Group writeups due in class on Thursday, February 9.