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A portion of a university-level math homework assignment for a complex variables course. It includes instructions for using substitutions and contour integrals to compute trigonometric integrals. The homework contains two problems, one involving a general real number 'a' and the other involving 'sin t'.
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Recall that we use the substitutions
cos t =
z +
z
sin t =
2 i
z −
z
dt =
dz
iz
to compute trigonometric integrals by transforming them into contour integrals on the coun-
terclockwise circle |z| = 1.
Question 1. For a general real number a with a 2 < 1, show that
∫ (^2) π
0
dt
1 + a cos t
2 π √ 1 − a^2
Question 2. Use residues to show that
∫ (^2) π
0
dt
2 + sin t
2 π √ 3
