In the previous section, we saw some solved examples on integration by partial fractions. In this section, we will see some solved examples which involve both substitution and partial fractions.
Solved Example 23.30
Find ∫[cosxsin2x−sinx]dx
Solution:
1. Put u = sin x. Then du/dx = cos x
⇒ cos x dx = du
• So we want:
∫[cosxsin2x−sinx]dx = ∫[1u2−u]du
• The numerator is a polynomial of degree zero. The denominator is also a polynomial of degree 2.
2. So it is a proper rational function. We can straight away start partial fraction decomposition
3. First we factorize the denominator. We get:
u2 − u = u(u−1)
♦ All factors are linear
♦ And all factors are distinct from one another.
4. So this is case I. We are able to write:
1u2−u=1u(u−1)=A1u + A2u−1
Where A1 and A2 are real numbers.
5. To find A1 and A2, we make denominators same on both sides:
1u(u−1)=A1u + A2u−1 = A1(u−1) + A2(u)u(u−1)
6. Since denominators are same on both sides, we can equate the numerators. We get:
1 = A1(u−1) + A2(u)
7. After equating the numerators, we can use suitable substitution.
♦ Put u = 1. We get: A2 = 1
♦ Put u = 0. We get: A1 = −1
8. Now the result in (4) becomes:
1u2−u=1u(u−1)=−1u + 1u−1
9. So the integration becomes easy. We get:
−log|u|+log|u−1| + C
• The reader may write all the steps involved in the integration process.
10. Substituting for u, we get:
−log|sinx|+log|sinx−1| + C
Solved Example 23.31
Find ∫[(3sinx−2)cosx5−cos2x−4sinx]dx
Solution:
1. Put u = sin x. Then du/dx = cos x
⇒ cos x dx = du
• So we want:
∫[(3sinx−2)cosx5−cos2x−4sinx]dx = ∫[(3sinx−2)cosx5−(1−sin2x)−4sinx]dx = ∫[3u−24+u2−4u]du
• The numerator is a polynomial of degree 1. The denominator is a polynomial of degree 2.
2. So it is a proper rational function. We can straight away start partial fraction decomposition
3. First we factorize the denominator. We get:
4 + u2 − 4u = (u−2)2
♦ All factors are linear
♦ And all factors are not distinct from one another.
4. So this is case II. We are able to write:
3u−24+u2−4u=3u−2(u−2)2=A1u−2 + A2(u−2)2
Where A1 and A2 are real numbers.
5. To find A1 and A2, we make denominators same on both sides:
3u−2(u−2)2=A1u−2 + A2(u−2)2=A1(u−2) + A2(u−2)2
6. Since denominators are same on both sides, we can equate the numerators. We get:
3u−2 = A1(u−2) + A2
7. After equating the numerators, we can use suitable substitution.
♦ Put u = 2. We get: 4 = A2
♦ Put u = 0. We get: −2 = −2 A1 + 4. So A1 = 3
8. Now the result in (4) becomes:
3u−24+u2−4u=3u−2 + 4(u−2)2
9. So the integration becomes easy. We get:
3log|u−2|−4u−2 + C
• The reader may write all the steps involved in the integration process.
10. Substituting for u, we get:
3log|sinx−2|−4sinx−2 + C
= 3log|sinx−2|+42−sinx + C
= 3log(2−sinx)+42−sinx + C
Since (2 - sin x) is always +ve.
Solved Example 23.32
Find ∫[cosx(1−sinx)(2−sinx)]dx
[Hint: Put sin x = u]
Solution:
1. Put u = sin x. Then du/dx = cos x
⇒ cos x dx = du
• So we want:
∫[cosx(1−sinx)(2−sinx)]dx = ∫[1(1−u)(2−u)]du
• The numerator is a polynomial of degree zero. The denominator is a polynomial of degree 2.
2. So it is a proper rational function. We can straight away start partial fraction decomposition
3. First we factorize the denominator. But it is already in the factorized form:
(1−u)(2−u)
♦ All factors are linear
♦ And all factors are distinct from one another.
4. So this is case I. We are able to write:
1(1−u)(2−u)=A11−u + A22−u
Where A1 and A2 are real numbers.
5. To find A1 and A2, we make denominators same on both sides:
1(1−u)(2−u)=A11−u + A22−u=A1(2−u) + A2(1−u)(1−u)(2−u)
6. Since denominators are same on both sides, we can equate the numerators. We get:
1 = A1(2−u) + A2(1−u)
7. After equating the numerators, we can use suitable substitution.
♦ Put u = 2. We get: A2 = −1
♦ Put u = 1. We get: A1 = 1
8. Now the result in (4) becomes:
1(1−u)(2−u)=11−u − 12−u
9. So the integration becomes easy. We get:
−log|1−u|+log|2−u| + C
• The reader may write all the steps involved in the integration process.
10. Substituting for u, we get:
−log|1−sinx|+log|2−sinx| + C
= log|2−sinx1−sinx| + C
Alternate method:
1. Put t = sin x. Then dt/dx = cos x
⇒ cos x dx = dt
• So we want:
∫[cosx(1−sinx)(2−sinx)]dx = ∫[dt(1−t)(2−t)] = ∫[dt2−3t+t2]
= ∫[dtt2−3t+2]
2. Recall how we analyzed formula VII: ∫[dtat2+bt+c] = 1a∫[dxu2 ± k2]
• In our present case, a = 1, b = −3 and c = 2
3. So we can calculate u and k2:
• u=t+b2a = t+−32(1) = (t−3/2)
• ±k2=ca−b24a2 = 21−(−3)24(1)2 = 2−94 = −(1/4)
4. So we want:
∫[dtt2−3x+2] = 1a∫[duu2 ±k2]
= 11∫[dt(t−3/2)2 − (1/2)2]
[Recall that, we put u = t + b/(2a). So du = dt]
5. This integration can be done as shown below:
(i) Put v = (t−3/2). Then dv/dt = 1, which gives dv = dt
• So we want:
∫[dt(t−3/2)2 − (1/2)2] = ∫[dvv2−(1/2)2]
(ii) We have formula I: ∫[dvv2−m2]=12mlog|v−mv+m|+C
• In our present case, m = 1/2
6. So we get:
∫[dvv2−(1/2)2]=12(1/2)log|v−(1/2)v+(1/2)|+C
=log|t−(3/2)−(1/2)t−(3/2)+(1/2)|+C=log|t−2t−1|+C=log|sinx−2sinx−1|+C
Solved Example 23.33
Find ∫[1ex−1]dx
[Hint: Put ex = u]
Solution:
1. Put u = ex. Then du/dx = ex
⇒ ex dx = du
• So we want:
∫[1ex−1]dx = ∫[exex(ex−1)]dx = ∫[1u(u−1)]du
• The numerator is a polynomial of degree zero. The denominator is a polynomial of degree 2.
2. So it is a proper rational function. We can straight away start partial fraction decomposition
3. First we factorize the denominator. But it is already in the factorized form:
u(u−1)
♦ All factors are linear
♦ And all factors are distinct from one another.
4. So this is case I. We are able to write:
1u(u−1)=A1u + A2u−1
Where A1 and A2 are real numbers.
5. To find A1 and A2, we make denominators same on both sides:
1u(u−1)=A1u + A2u−1=A1(u−1) + A2uu(u−1)
6. Since denominators are same on both sides, we can equate the numerators. We get:
1 = A1(u−1) + A2u
7. After equating the numerators, we can use suitable substitution.
♦ Put u = 1. We get: A2 = 1
♦ Put u = 0. We get: A1 = −1
8. Now the result in (4) becomes:
1u(u−1)=−1u + 1u−1
9. So the integration becomes easy. We get:
−log|u|+log|u−1| + C
• The reader may write all the steps involved in the integration process.
10. Substituting for u, we get:
−log|ex|+log|ex−1| + C
= log|ex−1ex| + C
Solved Example 23.34
Find ∫[1x(xn+1)]dx
[Hint: Multiply numerator and denominator by xn-1 and put xn = u]
Solution:
1. We have:
∫[1x(xn+1)]dx = ∫[xn−1xn−1x(xn+1)]dx = ∫[xn−1xn(xn+1)]dx = ∫[nxn−1nxn(xn+1)]dx
• Put u=xn. Then dudx = nxn−1
⇒ nxn−1dx = du
• So we want:
∫[1x(xn+1)]dx = ∫[nxn−1nxn(xn+1)]dx = ∫[1nu(u+1)]du
• The numerator is a polynomial of degree zero. The denominator is a polynomial of degree 2.
2. So it is a proper rational function. We can straight away start partial fraction decomposition
3. First we factorize the denominator. But it is already in the factorized form:
u(u+1)
♦ All factors are linear
♦ And all factors are distinct from one another.
4. So this is case I. We are able to write:
1u(u+1)=A1u + A2u+1
Where A1 and A2 are real numbers.
5. To find A1 and A2, we make denominators same on both sides:
1u(u+1)=A1u + A2u+1=A1(u+1) + A2uu(u+1)
6. Since denominators are same on both sides, we can equate the numerators. We get:
1 = A1(u+1) + A2u
7. After equating the numerators, we can use suitable substitution.
♦ Put u = −1. We get: A2 = −1
♦ Put u = 0. We get: A1 = 1
8. Now the result in (4) becomes:
1u(u+1)=1u − 1u+1
9. So the integration becomes easy. We get:
1n[log|u|−log|u+1| + C1]
• The reader may write all the steps involved in the integration process.
10. Substituting for u, we get:
1n[log|xn|−log|xn+1|] + C1n
= 1nlog|xnxn+1| + C
The link below gives a few more miscellaneous examples:
Exercise 23.5
We
have completed a discussion on integration by partial fraction decomposition.
In the next section, we will see integration by separation.
Previous
Contents
Copyright©2025 Higher secondary mathematics.blogspot.com