RD Sharma Class 12 Ex 19.4 Solutions Chapter 19 Indefinite Integrals

Here we provide RD Sharma Class 12 Ex 19.4 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 19.4 Solutions Chapter 19 Indefinite Integrals book pdf download. Now you will get step-by-step solutions to each question.

Textbook	NCERT
Class	Class 12th
Subject	Maths
Chapter	19
Exercise	19.4
Category	RD Sharma Solutions

Table of Contents

RD Sharma Class 12 Ex 19.4 Solutions Chapter 19 Indefinite Integrals

Question 1. Integrate $∫ \frac{(x^2 + 5x +2)}{(x + 2)} dx$

Solution:

Let, I = $∫ \frac{(x^2 + 5x +2)}{(x + 2)} dx$

Use division method, then we get,

$\frac{(x2 + 5x +2)}{(x+2)} = x + 3 - \frac{4}{(x+2)}$

$∫ \frac{(x^2 + 5x +2)}{(x + 2)} =∫ x + 3 - \frac{4}{(x+2)} dx$

= ∫ (x + 3)dx – 4∫1/(x + 2) dx

Integrate the above equation, then we get

= x²/2 + 3x – 4 log |x + 2| + c

Hence, I = x²/2 + 3x – 4 log |x + 2| + c

Question 2. Integrate $∫ \frac{x^3}{(x-2)} dx$

Solution:

Let I = $∫ \frac{x^3}{(x-2)} dx$

Use division method, then we get,

$\frac{x^3}{(x+2)} =x^2 + 2x + 4 + \frac{8}{(x-2)}$

= ∫ x² dx + 2∫x dx + 4∫ dx + 8 ∫1/(x – 2) dx

Integrate the above equation, then we get

= x³/3 + 2x²/2 + 4x + 8 log|x – 2| + c

= x³/3 + x² + 4x + 8 log|x – 2| + c

Hence, I = x³/3 + x² + 4x + 8 log|x – 2| + c

Question 3. Integrate $∫ \frac{(x^2 + x + 5)}{(3x +2)} dx$

Solution:

Let, I = $∫ \frac{(x^2 + x + 5)}{(3x +2)} dx$

Use division method, then we get,

$\frac{(x^2 + x + 5)}{(3x +2)} = \frac{x}{3} + \frac{1}{9} + \frac{43}{9}\times \frac{1}{3x+2} dx$

= $∫\frac{x}{3}dx + \frac{1}{9}∫1dx + \frac{43}{9} ∫\frac{1}{(3x+2)} dx$

Integrate the above equation, then we get

= $\frac{x^2}{6} + \frac{x}{9} + \frac{43}{9} \frac{1}{3} log|3x+2| + c$

= x² /6 + x/9 + (43/27) log|3x + 2| + c

Hence, I = = x² /6 + x/9 + (43/27) log|3x + 2| + c

Question 4. Integrate $∫ \frac{(2x+3)}{(x-1)^2} dx$

Solution:

Let, I = $∫ \frac{(2x+3)}{(x-1)^2} dx$
We can write the above equation as below,
= $∫ \frac{(2x + 2 - 2 + 3)}{(x-1)^2} dx$
On solving the above equation,
= $∫ \frac{(2x - 2 + 5)}{(x-1)^2} dx$
= $∫ \frac{2(x - 1)}{(x-1)^2} dx + 5 ∫\frac{1}{(x-1)^2} dx$
= 2∫ (1/(x – 1) dx + 5 ∫(x – 1)^-2 dx
Integrate the above equation, then we get
= 2 log|x – 1| + 5 (x – 1)^-1/(-1) + c
= 2 log|x – 1| – 5 / (x – 1) + c
Hence, I = 2 log|x – 1| – 5 / (x – 1) + c

Question 5. Integrate $∫ \frac{(x^2 + 3x - 1)}{(x+1)^2} dx$

Solution:

Let, I = $∫ \frac{(x^2 + 3x - 1)}{(x+1)^2} dx$
We can write the above equation as below,
= $∫\frac{(x^2 + x + 2x - 1)}{(x+1)^2} dx$
= $∫\frac{x(x+1)}{(x+1)^2} dx + ∫\frac{(2x - 1)}{(x+1)^2} dx$
= $∫\frac{x}{(x+1)} dx + ∫\frac{\sqrt{2x + 2 - 2 + 1}}{(x+1)^2} dx$
= $∫\frac{(x+1)}{(x+1)} dx - ∫\frac{1}{(x+1)} dx + 2∫\frac{(x + 1)}{(x+1)^2} dx - 3∫\frac{1}{(x+1)^2} dx$
= ∫ dx – ∫ 1/ (x + 1) dx + 2∫1/(x + 1) dx -3 ∫(x + 1)^-2 dx
Integrate the above equation, then we get
= x – log|x + 1| + 2 log|x + 1| – 3(x + 1)^-1/(-1) + c
= x – log|x + 1| + 2 log|x + 1| + 3/(x + 1) + c
= x + log|x + 1| + 3/(x + 1) + c
Hence, I = x + log|x + 1| + 3/(x + 1) + c

Question 6. Integrate $∫ \frac{(2x - 1)}{(x - 1)^2} dx$

Solution:

Let, I = $∫ \frac{(2x - 1)}{(x - 1)^2} dx$
= $∫ \frac{(2x - 1 + 2 - 2)}{(x - 1)^2} dx$
= $∫ \frac{(2x - 2 + 1)}{(x - 1)^2} dx$
= $∫ \frac{(2x - 2)}{(x - 1)^2} dx+ ∫\frac{1}{(x - 1)^2} dx$
= $2∫ \frac{(x - 1)}{(x - 1)^2} dx+ ∫\frac{1}{(x - 1)^2} dx$
= 2∫ 1/(x – 1) dx + ∫(x – 1)^-2 dx
Integrate the above equation, then we get
= 2 log|x – 1| + (x – 1)^-1/(-1) + c
= 2 log|x – 1| – 1/(x – 1) + c
Hence, I = 2 log|x – 1| – 1/(x – 1) + c

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RD Sharma Class 12 Ex 19.4 Solutions Chapter 19 Indefinite Integrals

Question 1. Integrate

Question 2. Integrate

Question 3. Integrate

Question 4. Integrate

Question 5. Integrate

Question 6. Integrate

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Question 1. Integrate $∫ \frac{(x^2 + 5x +2)}{(x + 2)} dx$

Question 2. Integrate $∫ \frac{x^3}{(x-2)} dx$

Question 3. Integrate $∫ \frac{(x^2 + x + 5)}{(3x +2)} dx$

Question 4. Integrate $∫ \frac{(2x+3)}{(x-1)^2} dx$

Question 5. Integrate $∫ \frac{(x^2 + 3x - 1)}{(x+1)^2} dx$

Question 6. Integrate $∫ \frac{(2x - 1)}{(x - 1)^2} dx$