72 NUMBER AND ALGEBRA
=[θ−θ^3
18+θ^5
600−θ^7
7(5040)+···] 10= 1 −1
18+1
600−1
7(5040)+···=0.946, correct to 3 significant figuresProblem 13. Evaluate∫ 0. 4
0 xln(1+x)dxusing
Maclaurin’s theorem, correct to 3 decimal
places.From Problem 4,
ln(1+x)=x−
x^2
2+x^3
3−x^4
4+x^5
5−···Hence∫ 0. 40xln(1+x)dx=∫ 0. 40x(x−x^2
2+x^3
3−x^4
4+x^5
5−···)dx=∫ 0. 40(x^2 −x^3
2+x^4
3−x^5
4+x^6
5−···)dx=[
x^3
3−x^4
8+x^5
15−x^6
24+x^7
35−···] 0. 40=(
(0.4)^3
3−(0.4)^4
8+(0.4)^5
15−(0.4)^6
24+(0.4)^7
35−···)
−(0)= 0. 02133 − 0. 0032 + 0. 0006827 −···
=0.019, correct to 3 decimal placesNow try the following exercise.
Exercise 37 Further problems on numerical
integration using Maclaurin’s series- Evaluate
∫ 0. 6
0. 2 3esinθdθ, correct to 3 decimal
places, using Maclaurin’s series. [1.784]- Use Maclaurin’s theorem to expand cos 2θ
and hence evaluate, correct to 2 decimal
places,∫ 10cos 2θθ1
3dθ. [0.88]- Determine the value of
∫ 1
0√
θcosθdθ, cor-
rect to 2 significant figures, using Maclaurin’s
series. [0.53]- Use√ Maclaurin’s theorem to expand
xln(x+1) as a power series. Hence
evaluate, correct to 3 decimal places,
∫ 0. 5
0
√
xln(x+1) dx. [0.061]8.6 Limiting values
It is sometimes necessary to find limits of the formlim
x→a{
f(x)
g(x)}
, wheref(a)=0 andg(a)=0.For example,lim
x→ 1{
x^2 + 3 x− 4
x^2 − 7 x+ 6}
=1 + 3 − 4
1 − 7 + 6=0
0and^00 is generally referred to as indeterminate.For certain limits a knowledge of series can some-
times help.
For example,lim
x→ 0{
tanx−x
x^3}≡lim
x→ 0⎧
⎪⎨⎪⎩x+1
3x^3 +···−xx^3⎫
⎪⎬⎪⎭from Problem 3=lim
x→ 0⎧
⎪⎨⎪⎩1
3x^3 +···x^3⎫
⎪⎬⎪⎭=lim
x→ 0{
1
3}
=1
3Similarly,lim
x→ 0{
sinhx
x}≡lim
x→ 0⎧
⎪⎪
⎨⎪⎪
⎩x+x^3
3!+x^5
5!+x⎫
⎪⎪
⎬⎪⎪
⎭from Problem 9=lim
x→ 0{
1 +x^2
3!+x^4
5!+···}
= 1However, a knowledge of series does not help withexamples such as lim
x→ 1{
x^2 + 3 x− 4
x^2 − 7 x+ 6}