Higher Engineering Mathematics, Sixth Edition

Integration by parts 425

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 thatan=0and

bn=−

2

n

cosnπ.

11. The equationC=

∫ 1

0

e−^0.^4 θcos1. 2 θdθ

and S=

∫ 1

0

e−^0.^4 θsin1. 2 θdθ

are involved in the study of damped

oscillations. Determine the values of C

andS.

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