RD Sharma Class 12 Ex 19.20 Solutions Chapter 19 Indefinite Integrals

Here we provide RD Sharma Class 12 Ex 19.20 Solutions Chapter 19 Indefinite Integrals for English medium students, Which will very helpful for every student in their exams. Students can download the latest Sharma Class 12 Ex 19.20 Solutions Chapter 19 Indefinite Integrals book pdf download. Now you will get step-by-step solutions to each question.

TextbookNCERT
ClassClass 12th
SubjectMaths
Chapter19
Exercise19.20
CategoryRD Sharma Solutions

Question 1. Evaluate: ∫(x+ x + 1)/(x– x) dx

Solution:

Given that I = ∫(x+ x + 1)/(x– x) dx

= ∫ [1 + (2x + 1)/(x– x)]dx

= x + ∫(2x + 1)/(x– x) dx + c

x + I+ c1           …….(i)

Now, I= ∫(2x + 1)/(x– x) dx

Let 2x + 1 = λ d/dx (x– x) + μ = λ(2x – 1) + μ

2x + 1 = (2λ)x – λ + μ

By comparing the coefficients of x, we get

2 = 2λ ⇒ λ = 1

-λ + μ = 1 ⇒ μ = 2

I= ∫ ((2x – 1) + 2)/(x– x) dx)

= ∫(2x – 1)/(x– x) dx + 2∫1/(x– x) dx

= ∫(2x – 1)/(x– x) dx + 2∫1/(x– 2x(1/2) + (1/2)– (1/2)2) dx

= ∫(2x – 1)/(x– x) dx + 2∫1/((x – 1/2)– (1/2)2) dx

= log⁡|x– x| + 2 × \frac{1}{2(\frac{1}{2})} log⁡|\frac{(x-\frac{1}{2}-\frac{1}{2})}{(x-\frac{1}{2}+\frac{1}{2})}|+c_1        

As we know that ∫1/(x– a2) dx = 1/2a log⁡|(x – a)/(x + a)| + c

So, I= log⁡|x– x| + 2log⁡|(x – 1)/x| + c2     ……(ii)

Now put the value of Iin eq(i), we get

I = x + log⁡|x– x| + 2log⁡|(x – 1)/x| + c

Question 2. ∫ (x+ x – 1)/(x+ x – 6) dx

Solution:

Given that I = ∫ (x+ x – 1)/(x+ x – 6) dx

= ∫[1 + 5 /(x+ x – 6)]dx

I = x + ∫5/(x+ x – 6) dx + c1   

Let us assume I= 5∫1/(x+ x – 6) dx

I = x + I+ c1     …..(i)

= 5∫ 1/(x+ 2x(1/2) + (1/2)– (1/2)– 6) dx

= 5∫ 1/((x + 1/2)– (5/2)2dx

= 5 1/2(5/2) log⁡|(x + 1/2 – 5/2)/(x + 1/2 + 5/2)| + c2  

As we know that ∫ 1/(x– a2) dx = 1/2a log⁡|(x – a)/(x + a)| + c

So, we get

I= log⁡|(x – 2)/(x + 3)| + c2      …….(ii)

Now put the value of Iin eq(i), we get

I = x + log⁡|(x – 2)/(x + 3)| + c

Question 3. ∫ (1 – x2)/(x(1 – 2x)) dx

Solution:

Given that I = ∫ (1 – x2)/(x(1 – 2x)) dx

= ∫ (1 – x2)/(x – 2x2) dx

= ∫ (x– 1)/(2x– x) dx

= ∫ [1/2 + (x/2 – 1)/(2x– x)]dx

I = 1/2x + ∫(x/2 – 1)/(2x– x) dx + c1

Let us assume I= ∫(x/2 – 1)/(2x– x) dx

So, I = 1/2x + I+ c1     …..(i)

Now, let x/2 -1 = λ d/dx (2x– x) + μ = λ(4x – 1) + μ

x/2 – 1 = (4λ)x – λ + μ

By comparing the coefficients of x, we get

1/2 = 4λ ⇒ λ = 1/8

-λ + μ = -1 ⇒ -(1/8) + μ = -1

μ = -7/8

I= ∫ (1/8(4x – 1) – 7/8)/(2x– x) dx

= 1/8 ∫(4x – 1)/(2x– x) dx – 7/8 ∫1/2(x– x/2)dx

= 1/8 ∫(4x – 1)/(2x– x) dx – 7/16 ∫1/(x– 2x(1/4) + (1/4)– (1/4)2) dx

= 1/8 ∫(4x – 1)/(2x– x) dx – 7/16 [1/((x – 1/4)– (1/4)2) dx

= 1/8 log⁡|2x– x| – 7/16 × 1/2(1/4) log⁡|(x – 1/4 – 1/4)/(x – 1/4 + 1/4)| + c2 

As we know that ∫ 1/(x– a2) dx = 1/2a log⁡|(x – a)/(x + a)| + c

So, we get

I= 1/8 log⁡|x| + 1/8 log⁡|2x – 1| – 7/8 log⁡|1 – 2x| + 7/8 log⁡2 + 7/8 log⁡|x| + c2   

I= log⁡|x| – 3/4 log⁡|1 – 2x| + c3       [Here, c= c+ 7/8 log⁡2]

Now put the value of Iin eq(i), we get

I = 1/2x + log⁡|x| – 3/4 log⁡|1 – 2x| + c

Question 4. ∫(x+ 1)/(x– 5x + 6)dx

Solution:

Given that I = ∫(x+ 1)/(x– 5x + 6)dx

Now we convert I into proper rational function by dividing x+ 1 by x– 5x + 6

So, 

(x+ 1)/(x– 5x + 6) = 1 + (5x – 5)/(x– 5x + 6) = 1 + (5x – 5)/((x – 2)(x – 3))

Let

(5x – 5)/((x – 2)(x – 3)) = A/(x – 2) + B/(x – 3)

So, we get A + B = 5 and 3A + 2B = 5

On solving both the equations we get A = -5 and B = 10 

So, \frac{(x^2 + 1)}{(x^2 - 5x + 6)} = 1 - \frac{5}{(x-2)} + \frac{10}{(x - 3)}

Hence, ∫(x+ 1)/((x + 1)2 (x + 3)) dx =∫dx – 5∫1/(x – 2) dx + 10∫x/(x – 3)

I = x – 5log⁡|x – 2| + 10log⁡|x – 3| + c

Question 5. ∫x2/(x+ 7x + 10) dx

Solution:

Given that I = ∫x2/(x+ 7x + 10) dx

= ∫ {1 – (7x + 10)/(x+ 7x + 10)}dx

I = x – ∫(7x + 10)/(x+ 7x + 10) dx + c1

Let us assume I= ∫(7x + 10)/(x+ 7x + 10) dx 

So, I = x – I+ c1     …..(i)

Now let us assume 7x + 10 = λ d/dx (x+ 7x + 10) + μ = λ(2x + 7) + μ

7x + 10 = (2λ)x + 7λ + μ

By comparing the coefficients of x, we get

7 = 2λ ⇒ λ = 7/2

7λ + μ = 10 ⇒ 7(7/2) + μ = 10μ = -29/2

 So, I= ∫(7/2(2x + 7) – 29/2)/(x+ 7x + 10) dx

= 7/2 ∫((2x + 7))/(x+ 7x + 10) dx – 29/2∫1/(x+ 2x(7/2) + (7/2)– (7/2)+ 10) dx

= 7/2 ∫(2x + 7)/(x+ 7x + 10) dx – 29/2 {1/((x + 7/2)– (3/2)2) dx

= 7/2 log⁡|x+ 7x + 10| – 29/2 × 1/2(3/2) log⁡|(x + 7/2 – 3/2)/(x + 7/2 + 3/2)| + c2 

As we know that ∫ 1/(x– a2) dx = 1/2a log⁡|(x – a)/(x + a)| + c

So, we get

I= 7/2 log⁡|x+ 7x + 10| – 29/6 log⁡|(x + 2)/(x + 5)| + c2

Now put the value of Iin eq(i), we get

I = x – 7/2 log⁡|x+ 7x + 10| + 29/6 log⁡|(x + 2)/(x + 5)| + c

Question 6. ∫(x+ x + 1)/(x– x + 1) dx

Solution:

Given that l = ∫(x+ x + 1)/(x– x + 1) dx

= ∫[1 + 2x/(x– x + 1)]dx

= x + ∫2x/(x– x + 1) dx + c1  

Let us assume I= ∫2x/(x– x + 1) dx

So, I = x + I+ c1     …..(i)

Now let 2x = λ d/dx (x– x + 1) + μ = λ(2x – 1) + μ

2x = (2λ) × -λ + μ

By comparing the coefficients of x, we get

2 = 2λ ⇒ λ = 1

-λ + μ = 0 ⇒ -1 + μ = 0

μ = 1

So, I= ∫((2x – 1) + 1)/(x– x + 1) dx

= ∫((2x – 1))/(x– x + 1) dx + ∫1/(x– 2x(1/2) + (1/2)– (1/2)+ 1) dx

= ∫(2x – 1)/(x– x + 1) dx + ∫1/((x – 1/2)+ (√3/2)2) dx

= log⁡|x– x + 1| + 2/√3 tan-1⁡((x – 1/2)/(√3/2)) + c2  

As we know that, ∫1/(x+ a2) dx = 1/a tan-1⁡(x/a) + c

So, we get

I= log⁡|x– x + 1| + 2/√3 tan-1⁡((2x – 1)/√3) + c2

Now put the value of Iin eq(i), we get

I = x + log⁡|x– x + 1| + 2/√3 tan-1⁡((2x – 1)/√3) + c

Question 7. ∫(x – 1)2/(x+ 2x + 2) dx

Solution:

Given that, I = ∫(x – 1)2/(x+ 2x + 2) dx

= ∫(x– 2x + 1)/(x+ 2x + 2) dx

= ∫[1 – (4x + 1)/(x+ 2x + 2)]dx

= x – ∫(4x + 1)/(x+ 2x + 2) dx + c1

Let us assume l1 = ∫(4x + 1)/(x+ 2x + 2) dx

So, I = x – I+ c1     …..(i)

Now, let 4x + 1 = λ d/dx (x+ 2x + 2) + μ

= λ(2x + 2) + μ = (2k)x + (2λ + μ)

By comparing the coefficients of x, we get

4 = 2λ ⇒ λ = 2

2λ + μ = 1 ⇒ 2(2) + μ = 1

μ = -3

l= ∫ (2(2x + 2) – 3)/(x+ 2x + 2) dx

= 2∫ ((2x + 2))/(x+ 2x + 2) dx – 3∫1/(x– 2x + (1)– (1)+ 2) dx

= 2∫ (2x + 2)/(x+ 2x + 2) dx – 3∫1/((x + 1)+ (1)2) dx

As we know that, ∫1/(x+ 1) dx = tan-1⁡x + c 

So, we get

l= 2log⁡|x+ 2x + 2| – 3tan-1⁡(x + 1) + c2                       

Now put the value of Iin eq(i), we get

I = x – 2log⁡|x+ 2x + 2| + 3tan-1⁡(x + 1) + c

Question 8. ∫(x+ x+ 2x + 1)/(x– x + 1) dx

Solution:

Given that, I = ∫(x+ x+ 2x + 1)/(x– x + 1) dx

= ∫[x + 2 + (3x – 1)/(x– x + 1)]dx

= x2/2 + 2x + ∫(3x. -1)/(x– x + 1) dx + c1

Let us assume l= ∫(3x – 1)/(x– x + 1) dx

So, I = x2/2 + 2x + I+ c1     …..(i)

Now, let 3x – 1 = λ d/dx (x– x + 1) + μ = λ(2x – 1) + μ

3x – 1 = (2λ)x – λ + μ

By comparing the coefficients of x, we get

3 = 2λ ⇒ λ = 3/2

-λ + μ = -1 ⇒ -(3/2) + μ = -1

μ = 1/2

So, I= ∫(3/2(2x – 1) + 1/2)/(x– x + 1) dx

= 3/2 ∫ ((2x – 1))/(x– x + 1) dx + 1/2 ∫1/(x– 2x(1/2) + (1/2)– (1/2)+ 1) dx

= 3/2 ∫ (2x – 1)/(x– x + 1) dx + 1/2 ∫1/((x + 1/2)+ (√3/2)2) dx

= 3/2 log⁡|x– x + 1| + 1/2 × 2/√3 tan-1⁡((x + 1/2)/(√3/2)) + c2 

As we know that, ∫1/(x+ a2) dx = 1/a tan-1(x/a) + c

So, we get

I= 3/2 log⁡|x– x + 1| + 1/√3 tan-1⁡((2x + 1)/√3) + c2

Now put the value of Iin eq(i), we get

I = x2/2 + 2x + 3/2 log⁡|x– x + 1| + 1/√3 tan-1⁡((2x + 1)/√3) + c

Question 9. ∫(x2 (x+ 4))/(x+ 4) dx

Solution:

Given that, I = ∫(x2 (x+ 4))/((x+ 4)) dx

= ∫ (x+ 4x2)/((x+ 4)) dx

= ∫ [x– 4x+ 20 – 80/(x+ 4)]dx

= x5/5 – (4x3)/3 + 20x – 80 ∫1/(x+ 4) dx + c1

Let us assume I= ∫1/(x+ 4) dx

So, I = x5/5 – (4x3)/3 + 20x – 80I+ c1     …..(i)

Now, I= ∫1/(x+ (2)2) dx

As we know that, ∫1/(x+ a2) dx = 1/a tan-1⁡(x/a) + c

So, we get

I= 1/2 tan-1⁡(x/2) + c

Now put the value of Iin eq(i), we get

I = x5/5 – (4x3)/3 + 20x – 80/2 tan-1⁡(x/2) + c

I = x5/5 – (4x3)/3 + 20x – 40tan-1(x/2) + c

Question 10. ∫ x2/(x+ 6x + 12) dx

Solution:

Given that, l = ∫ x2/(x+ 6x + 12) dx

= ∫ [1 – (6x + 12)/(x+ 6x + 12)]dx

= x – ∫(6x + 12)/(x+ 6x + 12) dx + c1

Let us assume I= ∫(6x + 12)/(x+ 6x + 12) dx

So, I = x – I+ c1     …..(i)

Now, let 6x + 12 = λ d/dx (x+ 6x + 12) + μ = λ(2x + 6) + μ

6x + 12 = (2λ)x + 6λ + μ

By comparing the coefficients of the power of x, we get

6 = 2λ ⇒ λ = 3

6λ + μ = 12

 6(3) + μ = 12

μ = -6

So, l= ∫(3(2x + 6) – 6)/(x+ 6x + 12) dx

=3∫ ((2x + 6))/(x+ 6x + 12) dx – 6∫1/(x+ 2x(3) + (3)– (3)+ 12) dx

= 3∫ (2x + 6)/(x+ 6x + 12) dx + 6∫1/((x + 3)+ (√3)2) dx

As we know that, ∫1/(x+ a2) dx = 1/2 tan-1⁡(x/a) + c

So, we get

I= 3log⁡|x+ 6x + 12| + 6/√3 tan-1⁡((x + 3)/√3) + c2                   

I= 3log⁡|x+ 6x + 12| + 2√3 tan-1⁡((x + 3)/√3) + c2

Now put the value of Iin eq(i), we get

l = x – 3log⁡|x+ 6x + 12| + 2√3 tan-1((x + 3)/√3) + c

I think you got complete solutions for this chapter. If You have any queries regarding this chapter, please comment in the below section our subject teacher will answer you. We tried our best to give complete solutions so you got good marks in your exam.

If these solutions have helped you, you can also share rdsharmasolutions.in to your friends.

Leave a Comment

Your email address will not be published.