410 INTEGRAL CALCULUSThus
∫
2 x+ 3
(x−2)^2dx≡∫ {
2
(x−2)+7
(x−2)^2}
dx=2ln(x−2)−7
(x−2)+c⎡⎣∫
7
(x−2)^2dxis determined using the algebraicsubstitutionu=(x−2) — see Chapter 39.⎤⎦Problem 6. Find∫
5 x^2 − 2 x− 19
(x+3)(x−1)^2dx.It was shown in Problem 6, page 21:5 x^2 − 2 x− 19
(x+3)(x−1)^2≡2
(x+3)+3
(x−1)−4
(x−1)^2Hence∫
5 x^2 − 2 x− 19
(x+3)(x−1)^2dx≡∫ {
2
(x+3)+3
(x−1)−4
(x−1)^2}
dx=2ln(x+3)+3ln(x−1)+4
(x−1)+corln{
(x+3)^2 (x−1)^3}
+4
(x−1)+cProblem 7. Evaluate
∫ 1− 23 x^2 + 16 x+ 15
(x+3)^3dx,correct to 4 significant figures.It was shown in Problem 7, page 22:3 x^2 + 16 x+ 15
(x+3)^3≡3
(x+3)−2
(x+3)^2−6
(x+3)^3Hence∫
3 x^2 + 16 x+ 15
(x+3)^3dx≡∫ 1− 2{
3
(x+3)−2
(x+3)^2−6
(x+3)^3}
dx=[
3ln(x+3)+2
(x+3)+3
(x+3)^2] 1− 2=(
3ln4+2
4+3
16)
−(
3ln1+2
1+3
1)=− 0. 1536 , correct to 4 significant figuresNow try the following exercise.Exercise 164 Further problems on integra-
tion using partial fractions with repeated
linear factorsIn Problems 1 and 2, integrate with respect
tox.1.∫
4 x− 3
(x+1)^2dx[
4ln(x+1)+7
(x+1)+c]2.∫
5 x^2 − 30 x+ 44
(x−2)^3dx⎡⎢
⎣5ln(x−2)+10
(x−2)
−2
(x−2)^2+c⎤⎥
⎦In Problems 3 and 4, evaluate the definite inte-
grals correct to 4 significant figures.3.∫ 21x^2 + 7 x+ 3
x^2 (x+3)[1.663]4.∫ 7618 + 21 x−x^2
(x−5)(x+2)^2dx [1.089]- Show that
∫ 10(
4 t^2 + 9 t+ 8
(t+2)(t+1)^2)
dt= 2 .546,correct to 4 significant figures.41.4 Worked problems on integration
using partial fractions with
quadratic factorsProblem 8. Find∫
3 + 6 x+ 4 x^2 − 2 x^3
x^2 (x^2 +3)dx.