Integration by parts 425
10. In determining a Fourier series to represent f (x) = x in the range −π to π, Fourier
coefficients are given by:
a n =
1
π
π
−π
x cos nx dx
and b n =
1
π
π
−π
x sin nx dx
where n is a positive integer. Show by
using integration by parts that a n = 0 and
b n =−
2
n
cos nπ.
11. The equation C =
1
0
e
−0.4θ cos 1.2θ dθ
and
S =
1
0
e
−0.4θ sin 1.2θ dθ
are involved in the study of damped
oscillations. Determine the values of C
and S.
[C = 0.66, S = 0.41]
10. In determining a Fourier series to represent f (x) = x in the range −π to π, Fourier
coefficients are given by:
a n =
1
π
π
−π
x cos nx dx
and b n =
1
π
π
−π
x sin nx dx
where n is a positive integer. Show by
using integration by parts that a n = 0 and
b n =−
2
n
cos nπ.
11. The equation C =
1
0
e
−0.4θ cos 1.2θ dθ
and
S =
1
0
e
−0.4θ sin 1.2θ dθ
are involved in the study of damped
oscillations. Determine the values of C
and S.
[C = 0.66, S = 0.41]