• Skip to main content
  • Skip to primary sidebar
  • Skip to footer
  • NCERT Solutions
    • NCERT Books Free Download
  • TS Grewal
    • TS Grewal Class 12 Accountancy Solutions
    • TS Grewal Class 11 Accountancy Solutions
  • CBSE Sample Papers
  • GSEB Solutions
  • Maharashtra Board
  • Kerala Board
    • Kerala Syllabus 9th Standard Physics Solutions Guide
    • Kerala Syllabus 9th Standard Biology Solutions Guide
  • Goa Board

CBSE Tuts

CBSE Maths notes, CBSE physics notes, CBSE chemistry notes

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4.

  • Integrals Class 12 Ex 7.1
  • Integrals Class 12 Ex 7.2
  • Integrals Class 12 Ex 7.3
  • Integrals Class 12 Ex 7.5
  • Integrals Class 12 Ex 7.6
  • Integrals Class 12 Ex 7.7
  • Integrals Class 12 Ex 7.8
  • Integrals Class 12 Ex 7.9
  • Integrals Class 12 Ex 7.10
  • Integrals Class 12 Ex 7.11
Board CBSE
Textbook NCERT
Class Class 12
Subject Maths
Chapter Chapter 7
Chapter Name Integrals
Exercise Ex 7.4
Number of Questions Solved 25
Category NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4

Integrate the functions in exercises 1 to 23

Ex 7.4 Class 12 Maths Question 1.
\frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 }
Solution:
Let x3 = t ⇒ 3x²dx = dt
\int { \frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 } dx } =\int { \frac { dt }{ { t }^{ 2 }+1 } } ={ tan }^{ -1 }t+c
= tan-1 (x3)+c

Ex 7.4 Class 12 Maths Question 2.
\frac { 1 }{ \sqrt { 1+{ 4x }^{ 2 } } }
Solution:
\frac { 1 }{ 2 } \int { \frac { dx }{ \sqrt { \frac { 1 }{ 4 } +{ x }^{ 2 } } } } =\frac { 1 }{ 2 } \int { \frac { dx }{ \sqrt { { \left( \frac { 1 }{ 2 } \right) }^{ 2 }+{ x }^{ 2 } } } }
=\frac { 1 }{ 2 } log\left| 2x+\sqrt { 1+{ 4x }^{ 2 } } \right| +c

Ex 7.4 Class 12 Maths Question 3.
\frac { 1 }{ \sqrt { { (2-x) }^{ 2 }+1 } }
Solution:
put (2-x)=t
so that -dx=dt
⇒ dx=-dt
\int { \frac { dx }{ \sqrt { { (2-x) }^{ 2 }+1 } } } =-\int { \frac { dt }{ \sqrt { { t }^{ 2 }+1 } } } =-log|t+\sqrt { { t }^{ 2 }+1 } |+c
=log\left| \frac { 1 }{ (2-x)+\sqrt { { x }^{ 2 }-4x+5 } } \right| +c

Ex 7.4 Class 12 Maths Question 4.
\frac { 1 }{ \sqrt { 9-{ 25x }^{ 2 } } }
Solution:
\int { \frac { dx }{ \sqrt { 9-{ 25x }^{ 2 } } } } =\frac { 1 }{ 5 } \int { \frac { dx }{ \sqrt { { \left( \frac { 3 }{ 5 } \right) }^{ 2 }-{ x }^{ 2 } } } }
=\frac { 1 }{ 5 } { sin }^{ -1 }\left( \frac { x }{ \frac { 3 }{ 5 } } \right) +c\quad =\frac { 1 }{ 5 } { sin }^{ -1 }\left( \frac { 5x }{ 3 } \right) +c

Ex 7.4 Class 12 Maths Question 5.
\frac { 3x }{ 1+{ 2x }^{ 4 } }
Solution:
Put x²=t,so that 2x dx=dt
⇒x dx = \frac { dt }{ 2 }
\therefore \int { \frac { 3x }{ 1+{ 2x }^{ 4 } } dx } =\frac { 1 }{ 2 } \int { \frac { dt }{ 1+{ 2t }^{ 2 } } } =\frac { 3 }{ 4 } \int { \frac { dt }{ { \left( \frac { 1 }{ \sqrt { 2 } } \right) }^{ 2 }+{ t }^{ 2 } } }
=\frac { 3 }{ 2\sqrt { 2 } } { tan }^{ -1 }(\sqrt { 2t } )+c\quad =\frac { 3 }{ 2\sqrt { 2 } } { tan }^{ -1 }(\sqrt { { 2x }^{ 2 } } )+c

Ex 7.4 Class 12 Maths Question 6.
\frac { { x }^{ 2 } }{ 1-{ x }^{ 6 } }
Solution:
put x3 = t,so that 3x²dx = dt
\int { \frac { { x }^{ 2 } }{ 1-{ x }^{ 6 } } dx } \quad =\frac { 1 }{ 3 } \int { \frac { dt }{ 1-{ t }^{ 2 } } \quad =\frac { 1 }{ 6 } log } \left| \frac { 1+t }{ 1-t } \right| +c
=\frac { 1 }{ 6 } log\left| \frac { 1+{ x }^{ 3 } }{ 1-{ x }^{ 3 } } \right| +c

Ex 7.4 Class 12 Maths Question 7.
\frac { x-1 }{ \sqrt { { x }^{ 2 }-1 } }
Solution:
I=\int { \frac { x-1 }{ \sqrt { { x }^{ 2 }-1 } } dx } -\int { \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } dx } ,I={ I }_{ 1 }-{ I }_{ 2 }
put x²-1 = t,so that 2x dx = dt
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q7.1

Ex 7.4 Class 12 Maths Question 8.
\frac { { x }^{ 2 } }{ \sqrt { { x }^{ 6 }+{ a }^{ 6 } } }
Solution:
put x3 = t
so that 3x2dx = dt
I=\frac { 1 }{ 3 } \int { \frac { dt }{ { t }^{ 2 }+{ { (a }^{ 3 }) }^{ 2 } } =\frac { 1 }{ 3 } log\left| t+\sqrt { { t }^{ 2 }+{ a }^{ 6 } } \right| +c }
=\frac { 1 }{ 3 } log|{ x }^{ 3 }+\sqrt { { a }^{ 6 }+{ x }^{ 6 } } |+c

Ex 7.4 Class 12 Maths Question 9.
\frac { { sec }^{ 2 }x }{ \sqrt { { tan }^{ 2 }x+4 } }
Solution:
let tanx = t
sec x²dx = dt
I=\int { \frac { dt }{ \sqrt { { t }^{ 2 }+{ (2) }^{ 2 } } } } =log|t+\sqrt { { t }^{ 2 }+4 } |+c
=log|tanx+\sqrt { { tan }^{ 2 }x+4 } |+c

Ex 7.4 Class 12 Maths Question 10.
\frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } }
Solution:
\int { \frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } } dx } =\int { \frac { dx }{ \sqrt { { (x+1) }^{ 2 }+1 } } }
=log|(x+1)+\sqrt { { x }^{ 2 }+2x+2 } |+c

Ex 7.4 Class 12 Maths Question 11.
\frac { 1 }{ { 9x }^{ 2 }+6x+5 }
Solution:
\int { \frac { 1 }{ { 9x }^{ 2 }+6x+5 } } =\frac { 1 }{ 9 } \int { \frac { dx }{ { \left( x+\frac { 1 }{ 3 } \right) }^{ 2 }{ +\left( \frac { 2 }{ 3 } \right) }^{ 2 } } }
=\frac { 1 }{ 6 } { tan }^{ -1 }\left( \frac { 3x+1 }{ 2 } \right) +c

Ex 7.4 Class 12 Maths Question 12.
\frac { 1 }{ \sqrt { 7-6x-{ x }^{ 2 } } }
Solution:
I=\int { \frac { dx }{ \sqrt { { 4 }^{ 2 }-{ (x+3) }^{ 2 } } } } \quad ={ sin }^{ -1 }\left( \frac { x+3 }{ 4 } \right) +c

Ex 7.4 Class 12 Maths Question 13.
\frac { 1 }{ \sqrt { (x-1)(x-2) } }
Solution:
\int { \frac { 1 }{ \sqrt { (x-1)(x-2) } } dx } =\int { \frac { dx }{ \sqrt { { \left( x-\frac { 3 }{ 2 } \right) }^{ 2 }-{ \left( \frac { 1 }{ 2 } \right) }^{ 2 } } } }
=log\left| x-\frac { 3 }{ 2 } +\sqrt { { x }^{ 2 }-3x+2 } \right| +c

Ex 7.4 Class 12 Maths Question 14.
\frac { 1 }{ \sqrt { 8+3x-{ x }^{ 2 } } }
Solution:
\int { \frac { dx }{ \sqrt { 8+3x-{ x }^{ 2 } } } } =\int { \frac { dx }{ \sqrt { 8-\left( { x }^{ 2 }-3x \right) } } }
=\int { \frac { dx }{ \sqrt { { \left( \frac { \sqrt { 41 } }{ 2 } \right) }^{ 2 }-{ \left( x-\frac { 3 }{ 2 } \right) }^{ 2 } } } } \quad ={ sin }^{ -1 }\left( \frac { 2x-3 }{ \sqrt { 41 } } \right) +c

Ex 7.4 Class 12 Maths Question 15.
\frac { 1 }{ \sqrt { (x-a)(x-b) } }
Solution:
\int { \frac { dx }{ \sqrt { (x-a)(x-b) } } } =\int { \frac { dx }{ { \left( x-\frac { a+b }{ 2 } \right) }^{ 2 }-{ \left( \frac { a-b }{ 2 } \right) }^{ 2 } } }
=log\left| \left( x-\frac { a+b }{ 2 } \right) +\sqrt { (x-a)(x-b) } \right| +c

Ex 7.4 Class 12 Maths Question 16.
\frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } }
Solution:
let\quad I=\int { \frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } } } dx
put 2x²+x-3=t
so that (4x+1)dx=dt
let\quad I=\int { \frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } } } dx
\therefore I=\int { \frac { dt }{ \sqrt { t } } } ={ 2t }^{ \frac { 1 }{ 2 } }+c\quad =2\sqrt { { 2x }^{ 2 }+x-3 } +c

Ex 7.4 Class 12 Maths Question 17.
\frac { x+2 }{ \sqrt { { x }^{ 2 }-1 } }
Solution:
\int { \frac { x+2 }{ \sqrt { { x }^{ 2 }-1 } } dx } \quad =\int { \frac { x }{ \sqrt { { x }^{ 2 }-1 } } dx } +\int { \frac { 2 }{ \sqrt { { x }^{ 2 }-1 } } dx }
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q17.1

Ex 7.4 Class 12 Maths Question 18.
\frac { 5x-2 }{ 1+2x+{ 3x }^{ 2 } }
Solution:
put 5x-2=A\frac { d }{ dx }(1+2x+3x²)+B
⇒ 6A=5, A=\frac { 5 }{ 6 }-2=2A+B, B=-\frac { 11 }{ 3 }
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q18.1

Ex 7.4 Class 12 Maths Question 19.
\frac { 6x+7 }{ \sqrt { (x-5)(x-4) } }
Solution:
\int { \frac { 6x+7 }{ \sqrt { (x-5)(x-4) } } dx } =\int { \frac { (6x+7)dx }{ \sqrt { { x }^{ 2 }-9x+20 } } }
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q19.1

Ex 7.4 Class 12 Maths Question 20.
\frac { x+2 }{ \sqrt { 4x-{ x }^{ 2 } } }
Solution:
I=\int { \frac { x-2 }{ \sqrt { 4-{ (x-2) }^{ 2 } } } dx+4\int { \frac { dx }{ \sqrt { 4-{ (x-2) }^{ 2 } } } } }
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q20.1

Ex 7.4 Class 12 Maths Question 21.
\frac { x+2 }{ \sqrt { { x }^{ 2 }+2x+3 } }
Solution:
I=\frac { 1 }{ 2 } \int { \frac { 2x+2 }{ \sqrt { { x }^{ 2 }+2x+3 } } dx }
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q21.1

Ex 7.4 Class 12 Maths Question 22.
\frac { x+3 }{ { x }^{ 2 }-2x-5 }
Solution:
I=\frac { 1 }{ 2 } \int { \frac { 2x-2 }{ { x }^{ 2 }-2x-5 } dx } +\int { \frac { dx }{ { x }^{ 2 }-2x-5 } }
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q22.1

Ex 7.4 Class 12 Maths Question 23.
\frac { 5x+3 }{ \sqrt { { x }^{ 2 }+4x+10 } }
Solution:
I=\int { \frac { \frac { 5 }{ 2 } (2x+4)+(3-10) }{ \sqrt { { x }^{ 2 }+4x+10 } } dx }
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q23.1

Ex 7.4 Class 12 Maths Question 24.
\int { \frac { dx }{ { x }^{ 2 }+2x+2 } equals }
(a) xtan-1(x+1)+c
(b) (x+1)tan-1x+c
(c) tan-1(x+1)+c
(d) tan-1x+c
Solution:
(b) let\quad I=\int { \frac { dx }{ { x }^{ 2 }+2x+2 } } =\int { \frac { dx }{ (x+1)^{ 2 }+1 } }
= (x+1)tan-1x+c

Ex 7.4 Class 12 Maths Question 25.
\int { \frac { dx }{ \sqrt { 9x-{ 4x }^{ 2 } } } equals }
(a) \frac { 1 }{ 9 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c
(b) \frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 8x-9 }{ 9 } \right) +c
(c) \frac { 1 }{ 3 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c
(d) { sin }^{ -1 }\left( \frac { 9x-8 }{ 9 } \right) +c
Solution:
(b) \int { \frac { dx }{ \sqrt { 9x-{ 4x }^{ 2 } } } } =\frac { 1 }{ 2 } \left[ \frac { dx }{ \sqrt { \left( \frac { 9 }{ 8 } \right) ^{ 2 }-\left[ { x }^{ 2 }-{ \frac { 9 }{ 4 } }x+\left( \frac { 9 }{ 8 } \right) ^{ 2 } \right] } } \right]
\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 8x-9 }{ 9 } \right) +c

We hope the NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 help you. If you have any query regarding NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4, drop a comment below and we will get back to you at the earliest.

Primary Sidebar

NCERT Exemplar problems With Solutions CBSE Previous Year Questions with Solutoins CBSE Sample Papers

Recent Posts

  • MCQ Questions for Class 6 Science with Answers PDF Download Chapter Wise
  • MCQ Questions for Class 7 Science with Answers PDF Download Chapter Wise
  • NCERT Solutions for Class 6 Sanskrit Ruchira Bhag 1
  • MCQ Questions for Class 8 Science with Answers PDF Download Chapter Wise
  • NCERT Solutions for Class 7 Maths Chapter 15 Visualising Solid Shapes InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 14 Symmetry InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 13 Exponents and Powers InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 12 Algebraic Expressions InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 9 Rational Numbers InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 8 Comparing Quantities InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles InText Questions
  • NCERT Solutions for Class 7 Maths Chapter 4 Simple Equations InText Questions

Footer

Maths NCERT Solutions

NCERT Solutions for Class 12 Maths
NCERT Solutions for Class 11 Maths
NCERT Solutions for Class 10 Maths
NCERT Solutions for Class 9 Maths
NCERT Solutions for Class 8 Maths
NCERT Solutions for Class 7 Maths
NCERT Solutions for Class 6 Maths

SCIENCE NCERT SOLUTIONS

NCERT Solutions for Class 12 Physics
NCERT Solutions for Class 12 Chemistry
NCERT Solutions for Class 11 Physics
NCERT Solutions for Class 11 Chemistry
NCERT Solutions for Class 10 Science
NCERT Solutions for Class 9 Science
NCERT Solutions for Class 7 Science
MCQ Questions NCERT Solutions
CBSE Sample Papers
NCERT Exemplar Solutions LCM and GCF Calculator
TS Grewal Accountancy Class 12 Solutions
TS Grewal Accountancy Class 11 Solutions