Trig Intervals Summary
1. SINES AND COSINES
(a) If sin(x) or cos(x) is to an odd power.
Factor out a term from the odd power. Use the identity sin2(x) + cos2(x) = 1.
Do a substitution (u = sin(x) or u = cos(x) as appropriate).
(b) If sin(x) and cos(x) both have even powers: Simplify with half-angle identities.
2. TANGENTS AND SECANTS
(a) If sec(x) has an even power.
Factor out sec2(x). Use the identity sec2(x) = tan2(x) + 1.
Do a substitution (u = tan(x)).
(b) If tan(x) has an odd power (and at least one sec(x)):
Factor out sec(x) tan(x). Use the identity tan2(x) = sec2(x)
Do a substitution (u = sec(x)).
Odd power on sine: Use u = cos(x).
sin3(x) cos2(x) dx = sin2(x) cos2(x) sin(x) dx = (1 - cos2(x)) cos2(x) sin(x) dx.
Odd power on cosine: Use u = sin(x).
sin4(x) cos3(x) dx = sin4(x) cos2(x) cos(x) dx = sin4(x)(1- sin2(x)) cos(x) dx.
Only even powers: Integrate directly as follows:
| Even power on secant: Use u = tan(x). tan2(x) sec4(x) dx = tan2(x) sec2(x) sec2(x) dx = tan2(x)(tan2(x) + 1) sec2(x) dx. Odd power on tangent: Use u = sec(x). tan3(x) sec2(x) dx = tan2(x) sec(x) sec(x) tan(x) dx = (sec2(x)-1) sec(x) sec(x) tan(x) dx. NOTES (a) For cot(x)/csc(x) the cases would be nearly identical to tan(x)/sec(x). (b) If you are stuck, try changing everything to sin(x) and cos(x) (or changing everything to sec(x) and tan(x). If you are still stuck, look at all your trig identities and rewrite the integral in another way. (c) And remember that we have added the following to our table of known integrals (use these, you don’t have to derive them): |