Class 12 RD Sharma Solutions – Chapter 19 Indefinite Integrals – Exercise 19.28

Last Updated : 11 Feb, 2021

Question 1. Find $\int\sqrt{3+2x-x^2}dx$

Solution:

$\int\sqrt{3+2x+x^2}dx=\int\sqrt{4-(x-1)^2}dx\\$

Let considered x – 1 = t,

so that dx = dt

Thus, $\int\sqrt{3+2x+x^2}dx\\=\int\sqrt{4-t^2}dt\\ =\frac{1}{2}t\sqrt{4-t^2}+\frac{4}{2}sin^{-1}\left(\frac{t}{2}\right)+C\\ =\frac{1}{2}(x-1)\sqrt{3+2x-x^2}+2sin^{-1}\left(\frac{x-1}{2}\right)+C$

Question 2. Evaluate $\int\sqrt{x^2+x+1}dx$

Solution:

Let I = $\int\sqrt{x^2+x+1}dx\\ =\int\sqrt{x^2+x+\frac{1}{4}+\frac{3}{4}}dx\\ =\int\sqrt{\left(x+\frac{1}{2}^2\right)+\left(\frac{\sqrt{3}}{2}\right)^2}dx\\ =\frac{\left(x+\frac{1}{2}\right)}{2}\sqrt{\left(x+\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}+\frac{\left(\frac{\sqrt{3}}{2}\right)^2}{2}.log\left|\left(x+\frac{1}{2}\right)+\sqrt{\left(x+\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}\right|+c\\ =\left(\frac{2x+1}{4}\right)\sqrt{x^2+x+1}+\frac{3}{8}log\left|\left(\frac{2x+1}{2}\right)+\frac{1}{2}\sqrt{x^2+x+1}\right|+c\\ I=\left(\frac{2x+1}{4}\right)\sqrt{x^2+x+1}+\frac{3}{8}log\left|{2x+1}+\sqrt{x^2+x+1}\right|+c\\$

Question 3. Evaluate $\int\sqrt{x-x^2}dx$

Solution:

I = $\int\sqrt{x-x^2}dx\\ \int\sqrt{\frac{1}{4}-\frac{1}{4}+x-x^2}dx\ \ \ \ \ \ -(Add\ and\ subtract \frac{1}{4})\\ \\ \int\sqrt{\left(\frac{1}{2}\right)^2-\left(\frac{1}{2}-x\right)^2}dx\\ = -\left(\frac{1-2x}{4}\right)\sqrt{\left(\frac{1}{2}\right)^2-\left(\frac{1}{2}-x\right)^2}-\frac{\left(\frac{1}{2}\right)^2}{2}sin^{-1}\left(\frac{\frac{1-x}{2}}{\frac{1}{2}}\right)+c\\$

Hence, $I=\left(\frac{2x-1}{4}\right)\sqrt{x-x^2}+\frac{1}{8}sin^{-1}(2x-1)+c$

Question 4. Evaluate $\int\sqrt{1+x-2x^2}dx$

Solution:

Let I = $\int\sqrt{1+x-2x^2}dx\\ =\sqrt{2}\int\sqrt{\frac{1}{2}+\frac{x}{2}-x^2}dx\\ =\sqrt{2}\int\sqrt{\frac{9}{16}+-\left(\frac{1}{16}-\frac{x}{2}+x\right)^2}dx\\ =\sqrt{2}\int\sqrt{\left(\frac{3}{4}\right)^2-\left(x-\frac{1}{4}\right)^2}dx\\ =\sqrt{2}\left\{\frac{\left(x-\frac{1}{4}\right)}{2}\sqrt{\frac{1}{2}+\frac{x}{2}-x^2}+\frac{9}{32}sin^{-1}\left(\frac{x-\frac{1}{4}}{\frac{3}{4}}\right)\right\}+c\\$

Therefore, I = $\frac{1}{8}(4x-1)\sqrt{1+x-2x^2}+\frac{9\sqrt{2}}{32}sin^{-1}\left(\frac{4x-1}{3}\right)+c$

Question 5. $\int cosx\sqrt{4-sin^2x}dx$

Solution:

I = $\int cosx\sqrt{4-sin^2x}dx$

Let us considered sinx = t

So, on differentiating, we get

cosx dx = dt

I = $\int\sqrt{4-t^2}dt\\ =\int\sqrt{2^2-t^2}dt\\ =\frac{t}{2}\sqrt{2^2-t^2}+\frac{4}{2}sin^{-1}\frac{t}{2}+c\\$

Therefore, I = $\frac{1}{2}sinx\sqrt{4-sin^2x}+2sin^{-1}\left(\frac{sinx}{2}\right)+c$

Question 6. Evaluate $\int e^x\sqrt{e^{2x}+1}dx$

Solution:

I = $\int e^x\sqrt{e^{2x}+1}dx$

Let us considered e^x = t

So, on differentiating, we get

e^xdx = dt

Therefore, I = $\int\sqrt{t^2+1^2}dt\\ =\frac{t}{2}\sqrt{t^2+1^2}+\frac{1}{2}log\left|t+\sqrt{t^2+1}\right|+c$

Hence, I = $\frac{e^x}{2}\sqrt{e^{2x}+1}+\frac{1}{2}log\left|e^x+\sqrt{e^{2x}+1}\right|+c$

Question 7. Evaluate $\int\sqrt{9-x^2}dx$

Solution:

I = $\int\sqrt{3^2-x^2}$

We already have,

$\int\sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}+c$

Therefore, I = $\frac{x}{2}\sqrt{9-x^2}+\frac{9}{2}sin^{-1}\frac{x}{3}+c$

Question 8. Evaluate $\int\sqrt{16x^2+25}dx$

Solution:

Let us assume I = $\int\sqrt{16x^2+25}dx$

$=\int\sqrt{(4x)^2+5^2}dx\\ =4\int\sqrt{x^2+\left(\frac{5}{4}\right)^2}dx\\ =4\left\{\frac{x}{2}\sqrt{x^2+\left(\frac{5}{4}\right)^2}+{\frac{\left(\frac{5}{4}\right)^2}{2}}log\left|x+\sqrt{x^2+\left(\frac{5}{4}\right)^2}\right|\right\}+c\\$

Therefore, I = $2x\sqrt{x^2+\frac{25}{16}}+\frac{25}{8}log\left|x+\sqrt{x^2+\frac{25}{16}}\right|+c\\$

Question 9. Evaluate $\int\sqrt{4x^2-5}dx$

Solution:

Let us assume I = $\int\sqrt{4x^2-5}dx$

$=2\int\sqrt{x^2-\left(\frac{\sqrt5}{2}\right)}dx\\ =2\left\{\frac{x}{2}\sqrt{x^2-\frac{5}{4}}-\frac{5}{8}log\left|x+\sqrt{x^2-\frac{5}{4}}\right|+c\right\}$

Therefore, I = $x\sqrt{x^2-\frac{5}{4}}-\frac{5}{4}log\left|x+\sqrt{x^2-\frac{5}{4}}\right|+c$

Question 10. Evaluate $\int\sqrt{2x^2+3x+4}dx$

Solution:

Let us assume I = $\int\sqrt{2x^2+3x+4}dx$

$=\sqrt{2}\int\sqrt{x^2+\frac{3}{2}x+2}dx\\ =\sqrt{2}\int\sqrt{x^2+\frac{3}{2}x+\frac{9}{16}+\frac{23}{16}}dx\\ =\sqrt{2}\int\sqrt{\left(x+\frac{3}{4}\right)^2+\left(\frac{\sqrt{23}}{4}\right)^2}dx\\ =\sqrt{2}\left\{\frac{\left(x+\frac{3}{4}\right)}{2}\sqrt{x^2+\frac{3}{2}x+2}+\frac{23}{32}.log\left|\left(x+\frac{3}{4}\right)+\sqrt{x^2+\frac{3}{2}x+2}\right|+c\right\}$

Therefore, I = $\frac{4x+3}{8}\sqrt{2x^2+3x+4}+\frac{23\sqrt{2}}{32}.log\left|\left(x+\frac{3}{4}\right)+\sqrt{x^2+\frac{3}{2}x+2}\right|+c$

Question 11. Evaluate $\int\sqrt{3-2x-2x^2}dx$

Solution:

Let us assume I = $\int\sqrt{3-2x-2x^2}dx$

$=\sqrt{2}\int\sqrt{\frac{3}{2}-x-x^2}dx\\ =\sqrt{2}\int\sqrt{\frac{7}{4}-\left(\frac{1}{4}+x+x^2\right)}dx\ \ \ \ -(Add\ and\ subtract\ \frac{1}{4})\\ =\sqrt{2}\int\sqrt{\left(\frac{\sqrt{7}}{2}\right)^2-\left(x+\frac{1}{2}\right)^2}dx\\ =\sqrt{2}\left\{\frac{x+\frac{1}{2}}{2}\sqrt{\frac{3}{2}-x+x^2}+\frac{7}{8}sin^{-1}\left(\frac{x+\frac{1}{2}}{\frac{\sqrt{7}}{2}}\right)+c\right\}$

Therefore, I = $\frac{2x+1}{4}\sqrt{3-2x-2x^2}+\frac{7\sqrt{2}}{8}sin^{-1}\left(\frac{2x+1}{\sqrt{7}}\right)+c$

Question 12. Evaluate $\int x\sqrt{x^4+1}dx$

Solution:

Let us assume x² = t

On differentiating we get

2x dx = dt

Therefore, I = $\frac{1}{2}\int\sqrt{t^2+1^2}dt\\ =\frac{1}{2}\left\{\frac{t}{2}\sqrt{t^2+1}+\frac{1}{2}log\left|t+\sqrt{t^2+1}\right|\right\}+c\\$

Hence, I = $\frac{1}{2}\left\{\frac{x^2}{2}\sqrt{x^4+1}+\frac{1}{2}log\left|x^2+\sqrt{x^4+1}\right|\right\}+c\\$

Question 13. Evaluate $\int x^2\sqrt{a^6-x^6}dx$

Solution:

I = $\int x^2\sqrt{a^6-x^6}dx$

Let us considered x³ = t

So, on differentiating, we get

3x²dx = dt

Therefore, I = $\frac{1}{3}\int\sqrt{a^6-t^2}dt\\ = \frac{1}{3}\left\{\frac{t}{2}\sqrt{a^6-t^2}+\frac{a^6}{2}sin^{-1}\left(\frac{t}{a^3}\right)\right\}+c\\$

Hence, I = $\frac{x^3}{6}\sqrt{a^6-x^6}+\frac{a^6}{6}sin^{-1}\left(\frac{x^3}{a^3}\right)+c$

Question 14. Evaluate $\int\sqrt{\frac{16+(logx)^2}{x}}dx$

Solution:

I = $\int\sqrt{\frac{16+(logx)^2}{x}}dx$

Let us considered logx = t

So, on differentiating, we get

1/x dx = dt

Therefore, I = $\int\sqrt{16+t^2}dt\\ =\int\sqrt{4^2+t^2}dt\\ =\frac{t}{2}\sqrt{16+t^2}+\frac{16}{2}log\left|t+\sqrt{16+t^2}\right|$

Hence, I = $\frac{logx}{2}\sqrt{16+(logx)^2}+8log\left|logx+\sqrt{16+(logx)^2}\right|+c$

Question 15. Evaluate $\int\sqrt{2ax-x^2}dx$

Solution:

I = $\int\sqrt{2ax-x^2}dx$

$=\int\sqrt{a^2-(a^2-ax+x^2)}dx\ \ \ \ \ -(Add\ and\ subtract\ a^2)\\ =\int\sqrt{a^2-(a-x)^2}dx\\ =\int\sqrt{a^2-(x-a)^2}dx\\ =\frac{(x-a)}{2}\sqrt{2ax-x^2}+\frac{a^2}{2}sin^{-1}\left(\frac{x-a}{a}\right)+c$

Therefore, I = $\frac{1}{2}(x-a)\sqrt{2ax-x^2}+\frac{a^2}{2}sin^{-1}\left(\frac{x-a}{a}\right)+c$

Question 16. Evaluate $\int\sqrt{3-x^2}dx$

Solution:

Let I = $\int\sqrt{3-x^2}dx$

$=\int\sqrt{(\sqrt{3})^2-x^2}dx$

I = $\frac{x}{2}\sqrt{3-x^2}+\frac{3}{2}sin^{-1}\left(\frac{x}{\sqrt{3}}\right)+c$

Suggest improvement

Class 12 RD Sharma Solutions - Chapter 19 Indefinite Integrals - Exercise 19.27

Class 12 RD Sharma Solutions - Chapter 19 Indefinite Integrals - Exercise 19.29

Share your thoughts in the comments

Chapter 1: Relations

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solutions of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors, and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

Chapter 1: Relations

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solutions of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors, and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

Class 12 RD Sharma Solutions – Chapter 19 Indefinite Integrals – Exercise 19.28

Question 1. Find

Question 2. Evaluate

Question 3. Evaluate

Question 4. Evaluate

Question 5.

Question 6. Evaluate

Question 7. Evaluate

Question 8. Evaluate

Question 9. Evaluate

Question 10. Evaluate

Question 11. Evaluate

Question 12. Evaluate

Question 13. Evaluate

Question 1. Find $\int\sqrt{3+2x-x^2}dx$

Question 2. Evaluate $\int\sqrt{x^2+x+1}dx$

Question 3. Evaluate $\int\sqrt{x-x^2}dx$

Question 4. Evaluate $\int\sqrt{1+x-2x^2}dx$

Question 5. $\int cosx\sqrt{4-sin^2x}dx$

Question 6. Evaluate $\int e^x\sqrt{e^{2x}+1}dx$

Question 7. Evaluate $\int\sqrt{9-x^2}dx$

Question 8. Evaluate $\int\sqrt{16x^2+25}dx$

Question 9. Evaluate $\int\sqrt{4x^2-5}dx$

Question 10. Evaluate $\int\sqrt{2x^2+3x+4}dx$

Question 11. Evaluate $\int\sqrt{3-2x-2x^2}dx$

Question 12. Evaluate $\int x\sqrt{x^4+1}dx$

Question 13. Evaluate $\int x^2\sqrt{a^6-x^6}dx$

Question 14. Evaluate $\int\sqrt{\frac{16+(logx)^2}{x}}dx$

Question 15. Evaluate $\int\sqrt{2ax-x^2}dx$

Question 16. Evaluate $\int\sqrt{3-x^2}dx$