Integration by Partial Fractions | Mathematics (Maths) Class 12 - JEE PDF Download

Introduction

We know that a Rational Number can be expressed in the form of p/q, where p and q are integers and q≠0. Similarly, a rational function is defined as the ratio of two polynomials which can be expressed in the form of partial fractions: P(x)/Q(x), where Q(x)≠0. 

1. Proper partial fraction: When the degree of the numerator is less than the degree of the denominator, then the partial fraction is known as a proper partial fraction.
2. Improper partial fraction: When the degree of the numerator is greater than the degree of denominator then the partial fraction is known as an improper partial fraction. Thus, the fraction can be simplified into simpler partial fractions, that can be easily integrated.

In this section, we show how to integer any rational function (a ratio of polynomials) by expressing it as a sum of simpler fractions, called partial fractions, that we already know how to integrate. To illustrate the method, observe that by taking the fractions 2/(x – 1) and 1/(x + 2) to a common denominator we obtain  

If we now reverse the procedure, we see how to integrate the function on the right side of this equation  
To see how the method of partial fractions works in general, let's consider a rational function  
Where P and Q are polynomials. It's possible to express f as sum of simpler fractions provided that the degree of P is less than the degree of Q. Such a rational function is called proper. Recall that if P(x) =
where an ≠ 0, then the degree of P is n and we write deg (P) = n.
If f is improper, that is, deg(P)  deg (Q), then we must take the preliminary step of dividing Q into P  (by division) until a remainder R(x) is obtained such that deg (R) < deg(Q). The division statement is

    where S and R are also polynomials. As the following example illustrates, sometimes this preliminary step is all that is required.

Ex.1 Evaluate dx.

Sol. Since the degree of the numerator is greater than the degree of the denominator, we first perform the long division. This enables us to write

The next step is to factor the denominator Q(x) as far as possible . It can be shown that any polynomial Q can be factored as a product of linear factors (of the form ax + b) and irreducible quadratic factors

(of the form ax² + bx + c, where b² – 4ac < 0).

For instance, if Q(x) = x⁴ – 16, we could factor it as Q(x) = (x² – 4) (x² + 4) = (x – 2) (x + 2) (x² + 4)
The third step is to express the proper rational function R(x)/Q(x) (from equation 1) as a sum of partial fractions of the form

A theorem in algebra guarantees that it is always possible to do this. We explain the details for the four cases that occur.

Case I : The Denominator Q(x) is a product of distinct linear factors.

This means that we can write Q(x) = (a₁x + b1) (a₂x + b2) ... (a_kx + bk)
where no factor is repeated (and no factor is a constant multiple of another). In this case the partial fraction theorem states that there exist constants A₁,A₂,...,A_k such that.

These constants can be determined as in the following example.

Ex.2 Evaluate  dx.

Sol. Since the degree of the numerator is less than the degree of the denominator, we don't need to divide.

We factor the denominator as 2x³ + 3x² – 2x = x(2x² + 3x – 2) = x(2x – 1) (x + 2)

Since the denominator has three distinct linear factors, the partial fraction decomposition of the integrand (2) has the form.

To denominator the values of A, B and C, we multiply both sides of this equation by the product of the denominators, x(2x – 1) (x + 2), obtaining.

(4) x² + 2x – 1 = A(2x – 1) (x + 2) + Bx(x + 2) + Cx(2x – 1)

Expanding the right side of equation 4 and writing it in the standard form for polynomials, we get

(5) x² + 2x – 1 = (2A + B + 2C) x² + (3A + 2B – C) x – 2A

The polynomials in Equation 5 are identical, so their coefficients must be equal. The coefficient of x² on the right side, 2A + B + 2C, must equal the coefficient of x² on the left side-namely, 1. Likewise. The coefficients of x are equal and the constant terms are equal. This gives the following system of equation for A, B and C.

2A + B + 2C = 1  , 3A + 2B – C = 2  , – 2A = –1

Solving, we get  

In integrating the middle term we have made the mental substitution u = 2x – 1, which gives du = 2dx and dx = du/2.

Case II : Q(x) is a product of linear factors, some of which are repeated

(6) 

By way of illustration, we could write  

but we prefer to work out in detail a simpler example.

Ex.3 Evaluate  dx

Sol.

The first step is to divide. The result of long division is  

The second step is to factor the denominator Q(x) = x³ – x² – x + 1. Since Q(1) = 0, we know that x – 1 is a factor and we obtain x³ – x² – x + 1 = (x – 1) (x² – 1) = (x – 1) (x – 1)(x + 1) = (x – 1)² (x + 1) Since the linear factor x – 1 occurs twice, the partial fraction decompositoin is

Multiplying by the least common denominator (x – 1)² (x + 1), we get

(7) 4x = A(x – 1)(x + 1) + B(x + 1) + C(x – 1)² = (A + C) x² + (B – 2x) x + (–A + B + C)

Now we equate coefficients : A + C = 0,B – 2C = 4, –A + B + C = 0

Solving, we obtain A = 1, B = 2, and C = – 1, so

Case III : Q(x) contains irreducible quadratic factors, none of which is repeated

(8)  where A and B are constants to be determined. For instance, the function givenby f(x) = x/[(x – 2)(x² + 1) (x² + 4)] has a partial fraction decomposition of the form

The term given in (8) can be integrated by completing the square and using the formula.

Ex.4 Evaluate  dx
Sol.

Since x³ + 4x = x(x² + 4) can't be factored further, we write  

Multiplying by x(x² + 4), we have 2x² – x + 4 = A (x² + 4) + (Bx + C) x = (A + B) x² + Cx + 4A Equating coefficients, we obtain A + B =2 C = –1 4A = 4

Thus A = 1, B = 1 and C = –1 and  

In order to integrate the second term we split it into to parts  

We make the substitution u = x² + 4 in the first of these integrals so that du = 2x dx. We evaluate the second integral by means of Formula 9 with a = 2.

Case IV : Q(x) Contains A repeated irreducible quadratic factor.

Ex.5 Write out the form of the partial fraction decomposition of the function

 
Sol.

Ex.6 Evaluate  

Sol. The form of the partial fraction decomposition is  

Multiplying by x(x² + 1)², we have –x³ + 2x² – x + 1 = A(x² + 1)² + (Bx + C) x (x² + 1) + (Dx + E)x

= A(x⁴ + 2x² + 1) + B(x⁴ + x²) + C(x³ + x) + Dx² + Ex = (A + B) x⁴ + Cx³ + (2A + B + D) x² + (C + E) x + A

If we equate coefficient, we get the system    A + B = 0 ,C = –1  ,2A + B + D = 2  ,C+E =–1 , A=1

Which the solution A = 1, B = –1, D = 1, and E = 0. Thus

We note that sometimes partial fractions can be avoided when integrating a rational function. For instance, although the integral  

could be evaluated by the method of case III, it's much easier to observe that if u = x(x² + 3) = x³ + 3x, then du = (3x² + 3) dx and so  

Ex.7 Evaluatedx

Sol. In this example there is a repeated quadratic polynomial in the denominator. Hence, according to our previous discussion

For some constants A₁, B₁, A₂ and B₂
An easy way to determine these constant is as follows. By long division,

Thus A₁ = 1, B₁ = – 3, A₂ = 1 and B₂ = 0

Ex.8 Evaluate  

Sol.

put sin x – cos x = s and sin x + cosx = t ⇒ (cosx + sinx) dx = ds and (cosx – sin x) dx = dt

Ex.9 Evaluate  

Sol.

We have   , [multiplying the Nr. and Dr. by e^x]

  putting e^x = t so that e^x dx = dt.

   1 ≡ A (t-1)2 + Bt(t-1)+Ct ....(1) (on resolving into partial fractions)

To find A, putting t = 0 on both sides of (1), we get A = 1

To find C, put t = 1 and we get C = 1. Thus 1 ≡ (t - 1)² + Bt (t - 1) + t

Comparing the coefficients of t² on both sides, we get 0 = 1 + B or B = – 1 

 = log t – log (t – 1) – {1/(t – 1)} + C

= log e^x – log (e^x – 1) – {1/(e^x – 1)} + c  = x – log (e^x – 1) – {1/(e^x – 1)} + c

Ex.10 Integrate (3x + 1) / {(x – 1)³ (x + 1)}.

Sol.

Putting x – 1 = y so that x = 1 + y, we get  

arranging the N^r. and the D^r. in ascending powers of y

Hence the required integral of the given fraction  

Ex.11 Evaluate the integral  

Sol.

Let x = sec²θ  ⇒   dx = 2 sec²θ tanq dθ

Ex.12 Integrate,  

Sol.

Ex.65 Integrate  

Sol. 

Put y = tanθ

Ex.13 Evaluate dx,  where f(x) is a polynomials of degree 2 in x such that  f(0) = f(1) = 3f(2) = –3. 

Sol.

Let , f(x) = ax² + bx + c   given, f(0) = f(1) = 3f(2) = –3

∴ f(0) = f(1) = 3f(2) = –3, f(0) = c = –3, f(1) = a + b + c = – 3,  3f(2) = 3 (4a + b + c) = – 3

on solving we get a = 1, b = – 1, c = –3  

∴  f(x) = x² – x – 3 

Using partical fractions, we get,  

We get, A = –1, B = 2, C = 2

Ex.14 Integrate 1/(sinx + sin 2x).

Sol.

Now putting cos x = t, so that – sin x dx = dt, we get