Ex. 7 p.335 u-Substitution and the Log Rule We can solve differential equations using the log rule as well. Solve the differential equation Solution - separate y things from x things and integrate both sides. Put the “plus C” on right side only. There are three basic choices for u: u = x, u = x ln x, and u = ln x. The first two don’t fit the u’/u pattern. If I rewrite the function to be the pattern fits because u = ln x and du = (1/x)dx Rewrite with u-substitution: Back-substitute:
Up until now, we didn’t know how to integrate tan x, cot x, sec x, and csc x. With the Log rule, we can now do integration of these functions. Ex 8 p. 336 Using a trig identity to integrate using log rule Find Rewrite with trig identity: Let u = cos x. Then du = – sin x dx
Ex 9 p. 336 Derivative of the secant function This problem needs a creative step, multiplying and dividing by the same quantity to make it work. Find Let u = sec x + tan x. (the denominator) Then du = sec x tan x + sec 2 x, which is the numerator! Integrate and back-substitute
The other two are left as problems in the homework. I remember these log ones by realizing if they are co- things, they have a negative in front. Secant and tangent go together, as do cosecant and cotangent. These can be written in different forms, see #83-86
Ex 10 p. 337 Integrating Trigonometric Functions Evaluate Using Pythagorean Identity, 1 + cot 2 x = csc 2 x Be careful with parentheses if graphing and getting definite integral with the calculator.
Last but not least, Ex 11 p. 337 Finding an average value Recall that average value refers to average height, which would be area divided by width. Find the average value of f(x) = tan x on the interval [0, π / 4]