Understanding Engineering Mathematics

These are called theorthogonality relations for sine and cosine. The limits−π,πon

the integrals may in fact be replaced byanyintegral of length 2π, or integer multiple

of 2π.

The coefficientsan,bnof the Fourier series for a given functionf(t)can be deter-

mined by multiplying through by cosnxor sinnx, integrating over a single period and

using the above orthogonality relations to remove all but the desired coefficient (see

Chapter 17). Show that in general

an=

1

π

∫π

−π

f(t)cosntdtn= 0 , 1 , 2 ,...

bn=

1

π

∫π

−π

f(t)sinntdtn= 1 , 2 ,...

Answers to reinforcement exercises

Note: an arbitrary constantCshould be added to each indefinite integral.

9.3.1 Definition of integration

A.(i) 9x^2 (ii)

√

3

2

√

x

(iii) −

8

x^5

(iv)

2

5

x−

3

5

(v) 3 cos 3x (vi) − 3 x^2 sinx^3 (vii) 5e^5 x (viii)

− 1

(x+ 1 )^2

(ix)

1

x

(x)

1

2

√

x+ 1

(xi)

3

3 x+ 1

(xii)

1

x^2 + 1

B.(i) 2 lnx (ii)

3

5

e^5 x (iii)

− 2

( 1 +x)

(iv) −

1

4 x^4

(v) sinx^3 (vi) −3tan−^1 x (vii)

5

2

x^2 /^5 (viii) 2

√

x

(ix)

2

3

sin 3x (x)

x^3

3

(xi) 6

√

x+ 1 (xii)

1

3

ln| 3 x+ 1 |

9.3.2 Standard integrals

A.(i) 4x (ii) x^3 −x^2 +x (iii)

− 3

4 x^4

(iv)

3

5

x^5 /^3

(v) sinx (vi) tanx (vii) xlnx−x (viii) tan−^1 x

B.(i) 2u^2 (ii)

2 s^3

3

−

3 s^2

2

+ 2 s (iii) −

1

t^6

(iv)

4

5

x^5 /^4

(v) −cosθ (vi) −cott (vii)

e^3 t

3

(viii) tan−^1 s