Higher Engineering Mathematics

INTEGRATION BY PARTS 423

H

10. In determining a Fourier series to repre-
    sentf(x)=xin the range−πtoπ, Fourier
    coefficients are given by:

an=

1

π

∫π

−π

xcosnxdx

and bn=

1

π

∫π

−π

xsinnxdx

where n is a positive integer. Show by

using integration by parts that an=0 and

bn=−

2

n

cosnπ.

11. The equationC=

∫ 1

0

e−^0.^4 θcos 1. 2 θdθ

and S=

∫ 1

0

e−^0.^4 θsin 1. 2 θdθ

are involved in the study of damped oscilla-

tions. Determine the values ofCandS.

[C= 0 .66,S= 0 .41]