Worksheet # 5: Trigonometric Integrals

1. What are the three Pythagorean trigonometric identities?

2. What are the power reduction (half angle) formulas for sin2 x and cos2 x?

3. Evaluate the following integrals.

(a)

∫cos2(x) dx

(b)

∫sin3 x dx

(c)

∫sin4(x) dx

(d)

∫ π/2

0

sin2 x cos2(x) dx

(e)

∫tan2 x dx

(f)

∫sin3 x cos2 x dx

(g)

∫x2 sinx dx

(h)

∫sinφ

cos3 φdφ

(i)

∫tan5 x sec3 x dx

(j)

∫sin 8x cos 5x dx (Hint: Use sum and difference formulas)

(k)

∫1− tan2 x

sec2 xdx

(l)

∫t sec2(t2) tan4(t2) dt

(m)

∫ π2

π4

cot3 x dx

(n)

∫sec3 x dx (Hint: Start out using integration by parts with u = secx)

4. Prove that for m,n integers

1

π

∫ 2π

0

cos (mx) cos (nx) =

{0 m 6= n1 m = n

.

5. Evaluate∫

sinx cosxdx by four methods:

(a) the substitution u = cosx

(b) the substitution u = sinx

(c) the identity sin 2x = 2 sinx cosx

(d) integration by parts