7.3 Calc Quiz

∫tan(x)dx

-ln|cosx|+C +ln|cosx|+C

∫cscxdx

-ln|cscx+cotx|+C

Solve: ∫tanxdx

=∫ sinx/cosx u=cosx du=-sinxdx -du=sinxdx =∫1/u *-du =-ln|u|+C =-ln|cosx|+C

∫ sin(^m)x*cos(^n)x dx What happens if m is any real number, and n is odd?

Split off cos(x), rewrite is as cos(^m-1)x in terms of sin(x) and then use u=sin(x)

∫ tan(^m)x*sec(^n)x dx What happens if m is odd?

Split off secxtanx, and then after that, rewrite the remaining tan(^m-1)x in terms of sec(x) using the identity 1+tan²x=sec²x. Use u=sec(x)

∫ tan(^m)x*sec(^n)x dx What happens if n is even?

Split off sec²x, rewrite sec(n-2)x in terms of tanx using the identity 1+tan²x=sec²x. Use u=tanx

∫ sin(^m)x*cos(^n)x dx What happens if m is odd, and n is any real number?

Split off the sin(x) and rewrite it as sin(^m-1)x in terms of cos(x) and use u=cos(x)

∫ sin(^m)x*cos(^n)x dx What happens if m and n are both non-negative, even integers?

Use the half angle identities to change the integral into a polynomial and then apply preceding strategies based on the powers of cos2x.

What about ∫ cot(^m)x*csc(^n)x dx

Use the same rules as ∫ tan(^m)x*sec(^n)x dx

∫sin^m (x)dx or ∫cos^m (m) dx What happens if m is odd?

Using the Pythagorean identity, cos(^2)x+sin(^2)x=1, split off a single factor of sin(x) or cos(x), which will give you cos/sin(^m-1), in terms of cos(x) or sin(x). Ex: ∫sin³xdx = ∫sin²xdx * ∫sinxdx = ∫((1-cos²x)(sinx)dx

∫sin^m (x)dx or ∫cos^m (m) dx What happens if m is even?

Using the half-angle identity, (1-cos2x)/2 = sin²x or (1+cos2x)/2 = cos²x, to break up the question.

∫ tan(^m)x*sec(^n)x dx What happens if m is even and n is odd?

We dont need to know.

∫sec(x)dx

ln|secx+tanx|+C

∫cot(x)dx

ln|sinx|+C