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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7.

Board CBSE
Textbook NCERT
Class Class 12
Subject Maths
Chapter Chapter 7
Chapter Name Integrals
Exercise Ex 7.7
Number of Questions Solved 11
Category NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7

Integral the function in exercises 1 to 9

Ex 7.7 Class 12 Maths Question 1.
\(\sqrt { 4-{ x }^{ 2 } } \)
Solution:
\(let\quad I=\int { \sqrt { 4-{ x }^{ 2 } } } dx=\int { \sqrt { { (2) }^{ 2 }-{ x }^{ 2 } } dx } \)
\(=\frac { x\sqrt { 4-{ x }^{ 2 } } }{ 2 } +2{ sin }^{ -1 }\left( \frac { x }{ 2 } \right) +c \)

Ex 7.7 Class 12 Maths Question 2.
\(\sqrt { 1-{ 4x }^{ 2 } } \)
Solution:
\(\int { \sqrt { 1-{ 4x }^{ 2 } } } dx=2\int { \sqrt { { \left( \frac { 1 }{ 2 } \right) }^{ 2 }-{ x }^{ 2 } } } dx\)
\(=\frac { x\sqrt { 1-{ 4x }^{ 2 } } }{ 2 } +\frac { 1 }{ 4 } { sin }^{ -1 }(2x)+c\)

Ex 7.7 Class 12 Maths Question 3.
\(\sqrt { { x }^{ 2 }+4x+6 } \)
Solution:
\(\int { \sqrt { { x }^{ 2 }+4x+6 } } dx=\int { \sqrt { { (x+2) }^{ 2 }+{ (\sqrt { 2 } ) }^{ 2 } } } dx\)
\(=\frac { x+2 }{ 2 } \sqrt { { x }^{ 2 }+4x+6 } +log\left| (x+2)+\sqrt { { x }^{ 2 }+4x+6 } \right| +c\)

Ex 7.7 Class 12 Maths Question 4.
\(\sqrt { { x }^{ 2 }+4x+1 } \)
Solution:
\(\int { \sqrt { { x }^{ 2 }+4x+1 } } dx=\int { \sqrt { { (x+2) }^{ 2 }-{ (\sqrt { 3 } ) }^{ 2 } } } dx\)
\(=\frac { x+2 }{ 2 } \sqrt { { x }^{ 2 }+4x+1 } -\frac { 3 }{ 2 } log\left| (x+2)+\sqrt { { x }^{ 2 }+4x+1 } \right| +c\)

Ex 7.7 Class 12 Maths Question 5.
\(\sqrt { 1-4x-{ x }^{ 2 } } \)
Solution:
\(\int { \sqrt { 1-4x-{ x }^{ 2 } } } dx=\int { \sqrt { { (5) }^{ 2 }-{ (x+2) }^{ 2 } } dx } \)
\(=\frac { x+2 }{ 2 } \sqrt { 5-{ (x+2) }^{ 2 } } dx \)

Ex 7.7 Class 12 Maths Question 6.
\(\sqrt { { x }^{ 2 }+4x-5 } \)
Solution:
\(\int { \sqrt { { x }^{ 2 }+4x-5 } } dx=\int { \sqrt { { (x+2) }^{ 2 }-{ (3) }^{ 2 } } } dx\)
\(=\frac { x+2 }{ 2 } \sqrt { { x }^{ 2 }+4x-5 } -\frac { 9 }{ 2 } log|x+2+\sqrt { { x }^{ 2 }+4x-5 } |+c\)

Ex 7.7 Class 12 Maths Question 7.
\(\sqrt { 1+3x-{ x }^{ 2 } } \)
Solution:
\(\int { \sqrt { 1-\left( { x }^{ 2 }-3x \right) } } dx\)
\(=\int { \sqrt { { \left( \frac { \sqrt { 13 } }{ 2 } \right) }^{ 2 }-{ \left( x-\frac { 3 }{ 2 } \right) }^{ 2 } } } dx\)
\(=\frac { 2x-3 }{ 4 } \sqrt { 1+3x-{ x }^{ 2 } } +\frac { 13 }{ 8 } { sin }^{ -1 }\left[ \frac { 2x-3 }{ \sqrt { 3 } } \right] +c\)

Ex 7.7 Class 12 Maths Question 8.
\(\sqrt { { x }^{ 2 }+3x } \)
Solution:
\(\int { \sqrt { { x }^{ 2 }+3x } } dx=\int { \sqrt { { \left( x+\frac { 3 }{ 2 } \right) }^{ 2 }-{ \left( \frac { 3 }{ 2 } \right) }^{ 2 } } } dx\)
\(=\frac { 2x+3 }{ 4 } \sqrt { { x }^{ 2 }+3x } -\frac { 9 }{ 8 } log\left| x+\frac { 3 }{ 2 } +\sqrt { { x }^{ 2 }+3x } \right| +c\)

Ex 7.7 Class 12 Maths Question 9.
\(\sqrt { 1+\frac { { x }^{ 2 } }{ 9 } } \)
Solution:
\(\int { \sqrt { 1+\frac { { x }^{ 2 } }{ 9 } } } dx=\frac { 1 }{ 3 } \int { \sqrt { { x }^{ 2 }+{ 3 }^{ 2 } } } \)
\(=\frac { 1 }{ 6 } \left[ x\sqrt { { x }^{ 2 }+9 } +9log|x+\sqrt { { x }^{ 2 }+9 } | \right] +c\)

Choose the correct answer in the Exercises 10 to 11:

Ex 7.7 Class 12 Maths Question 10.
\(\int { \sqrt { 1+{ x }^{ 2 } } } dx\quad is\quad equal\quad to\)
(a) \(\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log|x+\sqrt { 1+{ x }^{ 2 } } |+c\)
(b) \(\frac { 2 }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c\)
(c) \(\frac { 2 }{ 3 } x{ \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c\)
(d) \(\frac { { x }^{ 2 } }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } { x }^{ 2 }log\left| x+\sqrt { 1+{ x }^{ 2 } } \right| +c\)
Solution:
(a) \(\int { \sqrt { 1+{ x }^{ 2 } } } dx\)
\(=\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log(x+\sqrt { 1+{ x }^{ 2 } } )+c\)

Ex 7.7 Class 12 Maths Question 11.
\(\int { \sqrt { { x }^{ 2 }-8x+7 } } dx\quad is\quad equal\quad to\)
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7 Q11.1
Solution:
(d) \(\int { \sqrt { { x }^{ 2 }-8x+7 } } dx=\int { \sqrt { { (x-4) }^{ 2 }-{ (3) }^{ 2 } } } dx\)
\(=\frac { x-4 }{ 2 } \sqrt { { x }^{ 2 }-8x+7 } -\frac { 9 }{ 2 } log\left| (x-4)+\sqrt { { x }^{ 2 }+8x+7 } \right| +c\)

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