NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7

Exercise 7.7

1. Integrate $\sqrt {4 - {x^2}} $.

Ans: Let $I = \int {\sqrt {4 - {x^2}} dx = \int {\sqrt {{{\left( 2 \right)}^2} - {{\left( x \right)}^2}} dx} } $

It is known that,

$\sqrt {{a^2} - {x^2}} dx = \dfrac{x}{2}\sqrt {{a^2} - {x^2}}  + \dfrac{{{a^2}}}{2}{\sin ^{ - 1}}\dfrac{x}{a} + C$ 

For $a$ substitute $2$ and solve the integral.

$\therefore I = \dfrac{x}{2}\sqrt {4 - {x^2}}  + \dfrac{4}{2}{\sin ^{ - 1}}\dfrac{x}{2} + C$ 

$ = \dfrac{x}{2}\sqrt {4 - {x^2}}  + 2{\sin ^{ - 1}}\dfrac{x}{2} + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {4 - {x^2}} $ is $\dfrac{x}{2}\sqrt {4 - {x^2}}  + 2{\sin ^{ - 1}}\dfrac{x}{2} + C$.

2. Integrate $\sqrt {1 - 4{x^2}} $.

Ans: Let $I = \int {\sqrt {1 - 4{x^2}} dx = \int {\sqrt {{{\left( 1 \right)}^2} - {{\left( {2x} \right)}^2}} dx} } $

Let $2x = t \Rightarrow 2dx = dt$ 

$\therefore I = \dfrac{1}{2}\int {\sqrt {{{\left( 1 \right)}^2} - {{\left( t \right)}^2}} } $ 

It is known that,

$\sqrt {{a^2} - {x^2}} dx = \dfrac{x}{2}\sqrt {{a^2} - {x^2}}  + \dfrac{{{a^2}}}{2}{\sin ^{ - 1}}\dfrac{x}{a} + C$ 

For $a$ substitute $1$ then substitute $2x$ for $t$ and solve the integral.

$\therefore I = \dfrac{1}{2}\left[ {\dfrac{t}{2}\sqrt {1 - {t^2}}  + \dfrac{1}{2}{{\sin }^{ - 1}}t} \right] + C$

$ = \dfrac{t}{4}\sqrt {1 - {t^2}}  + \dfrac{1}{4}{\sin ^{ - 1}}t + C$

$ = \dfrac{{2x}}{4}\sqrt {1 - 4{x^2}}  + \dfrac{1}{4}{\sin ^{ - 1}}2x + C$

$ = \dfrac{x}{2}\sqrt {1 - 4{x^2}}  + \dfrac{1}{4}{\sin ^{ - 1}}2x + C$ 

Where $C$ is an arbitrary constant.

The integral of $\sqrt {1 - 4{x^2}} $ is $\dfrac{x}{2}\sqrt {1 - 4{x^2}}  + \dfrac{1}{4}{\sin ^{ - 1}}2x + C$.

3. Integrate $\sqrt {{x^2} + 4x + 6} $.

Ans: Let $I = \int {\sqrt {{x^2} + 4x + 6} dx} $

$I = \int {\sqrt {{x^2} + 4x + 4 + 2} dx} $

$I = \int {\sqrt {\left( {{x^2} + 4x + 4} \right) + 2} dx} $

$I = \int {\sqrt {{{\left( {x + 2} \right)}^2} + {{\left( {\sqrt 2 } \right)}^2}} dx} $

It is known that,

$\sqrt {{x^2} + {a^2}} dx = \dfrac{x}{2}\sqrt {{x^2} + {a^2}}  + \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} + {a^2}} } \right| + C$ 

For $a$ substitute $\sqrt 2 $ and for $x$ put $x + 2$ and solve the integral.

$\therefore I = \dfrac{{\left( {x + 2} \right)}}{2}\sqrt {{x^2} + 4x + 6}  + \dfrac{2}{2}\log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x + 6} } \right| + C$

$ = \dfrac{{\left( {x + 2} \right)}}{2}\sqrt {{x^2} + 4x + 6}  + \log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x + 6} } \right| + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {{x^2} + 4x + 6} $ is $\dfrac{{\left( {x + 2} \right)}}{2}\sqrt {{x^2} + 4x + 6}  + \log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x + 6} } \right| + C$.

4. Integrate $\sqrt {{x^2} + 4x + 1} $.

Ans: Let $I = \int {\sqrt {{x^2} + 4x + 1} dx} $

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

$I = \int {\sqrt {{{\left( {x + 2} \right)}^2} - {{\left( {\sqrt 3 } \right)}^2}} dx} $

It is known that,

$\sqrt {{x^2} - {a^2}} dx = \dfrac{x}{2}\sqrt {{x^2} - {a^2}}  - \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} - {a^2}} } \right| + C$ 

For $a$ substitute $\sqrt 3 $ and for $x$ put $x + 2$ and solve the integral.

$\therefore I = \dfrac{{\left( {x + 2} \right)}}{2}\sqrt {{x^2} + 4x + 1}  - \dfrac{3}{2}\log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x + 1} } \right| + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {{x^2} + 4x + 1} $ is $\dfrac{{\left( {x + 2} \right)}}{2}\sqrt {{x^2} + 4x + 1}  - \dfrac{3}{2}\log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x + 1} } \right| + C$.

5. Integrate $\sqrt {1 - 4x - {x^2}} $.

Ans: Let $I = \int {\sqrt {1 - 4x - {x^2}} dx} $

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

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

$I = \int {\sqrt {{{\left( {\sqrt 5 } \right)}^2} - {{\left( {x + 2} \right)}^2}} dx} $

It is known that,

$\sqrt {{a^2} - {x^2}} dx = \dfrac{x}{2}\sqrt {{a^2} - {x^2}}  + \dfrac{{{a^2}}}{2}{\sin ^{ - 1}}\dfrac{x}{a} + C$ 

For $a$ substitute $\sqrt 5 $ and for $x$ put $x + 2$ and solve the integral.

$\therefore I = \dfrac{{\left( {x + 2} \right)}}{2}\sqrt {1 - 4x - {x^2}}  + \dfrac{5}{2}{\sin ^{ - 1}}\left( {\dfrac{{x + 2}}{{\sqrt 5 }}} \right) + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {1 - 4x - {x^2}} $ is $\dfrac{{\left( {x + 2} \right)}}{2}\sqrt {1 - 4x - {x^2}}  + \dfrac{5}{2}{\sin ^{ - 1}}\left( {\dfrac{{x + 2}}{{\sqrt 5 }}} \right) + C$.

6. Integrate $\sqrt {{x^2} + 4x - 5} $.

Ans: Let $I = \int {\sqrt {{x^2} + 4x - 5} dx} $

$I = \int {\sqrt {\left( {{x^2} + 4x + 4} \right) - 9} dx} $

$I = \int {\sqrt {{{\left( {x + 2} \right)}^2} - {{\left( 3 \right)}^2}} dx} $

It is known that,

$\sqrt {{x^2} - {a^2}} dx = \dfrac{x}{2}\sqrt {{x^2} - {a^2}}  - \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} - {a^2}} } \right| + C$ 

For $a$ substitute $\sqrt 5 $ and for $x$ put $x + 2$ and solve the integral.

$\therefore I = \dfrac{{\left( {x + 2} \right)}}{2}\sqrt {{x^2} + 4x - 5}  - \dfrac{9}{2}\log \left| {\left( {x + 2} \right) + \sqrt {{x^2} + 4x - 5} } \right| + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {{x^2} + 4x - 5} $ is $\dfrac{{\left( {x + 2} \right)}}{2}\sqrt {1 - 4x - {x^2}}  + \dfrac{5}{2}{\sin ^{ - 1}}\left( {\dfrac{{x + 2}}{{\sqrt 5 }}} \right) + C$.

7. Integrate $\sqrt {1 + 3x - {x^2}} $.

Ans: Let $I = \int {\sqrt {1 + 3x - {x^2}} dx} $

$I = \int {\sqrt {1 - \left( {{x^2} - 3x + \dfrac{9}{4} - \dfrac{9}{4}} \right)} dx} $

$I = \int {\sqrt {\left( {1 + \dfrac{9}{4}} \right) - {{\left( {x - \dfrac{3}{2}} \right)}^2}} dx} $

$I = \int {\sqrt {{{\left( {\dfrac{{\sqrt {13} }}{2}} \right)}^2} - {{\left( {x - \dfrac{3}{2}} \right)}^2}} dx} $

It is known that,

$\sqrt {{a^2} - {x^2}} dx = \dfrac{x}{2}\sqrt {{a^2} - {x^2}}  + \dfrac{{{a^2}}}{2}{\sin ^{ - 1}}\dfrac{x}{a} + C$ 

For $a$ substitute $\dfrac{{\sqrt {13} }}{2}$ and for $x$ put $x - \dfrac{3}{2}$ and solve the integral.

$\therefore I = \dfrac{{\left( {x - \dfrac{3}{2}} \right)}}{2}\sqrt {1 - 3x - {x^2}}  + \dfrac{{13}}{{4 \times 2}}{\sin ^{ - 1}}\left( {\dfrac{{x - \dfrac{3}{2}}}{{\dfrac{{\sqrt {13} }}{2}}}} \right) + C$

$ = \dfrac{{2x - 3}}{4}\sqrt {1 - 3x - {x^2}}  + \dfrac{{13}}{8}{\sin ^{ - 1}}\left( {\dfrac{{2x - 3}}{{\sqrt {13} }}} \right) + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {1 + 3x - {x^2}} $ is $\dfrac{{2x - 3}}{4}\sqrt {1 - 3x - {x^2}}  + \dfrac{{13}}{8}{\sin ^{ - 1}}\left( {\dfrac{{2x - 3}}{{\sqrt {13} }}} \right) + C$.

8. Integrate $\sqrt {{x^2} + 3x} $.

Ans: Let $I = \int {\sqrt {{x^2} + 3x} dx} $

$I = \int {\sqrt {{x^2} + 3x + \dfrac{9}{4} - \dfrac{9}{4}} dx} $

$I = \int {\sqrt {{{\left( {x + \dfrac{3}{2}} \right)}^2} - {{\left( {\dfrac{3}{2}} \right)}^2}} dx} $

It is known that,

$\sqrt {{x^2} - {a^2}} dx = \dfrac{x}{2}\sqrt {{x^2} - {a^2}}  - \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} - {a^2}} } \right| + C$ 

For $a$ substitute $\dfrac{3}{2}$ and for $x$ put $x + \dfrac{3}{2}$ and solve the integral.

$\therefore I = \dfrac{{x + \dfrac{3}{2}}}{2}\sqrt {{x^2} + 3x}  - \dfrac{{\dfrac{9}{4}}}{2}\log \left| {\left( {x + \dfrac{3}{2}} \right) + \sqrt {{x^2} + 3x} } \right| + C$

$ = \dfrac{{\left( {2x + 3} \right)}}{2}\sqrt {{x^2} + 3x}  - \dfrac{9}{8}\log \left| {\left( {x + \dfrac{3}{2}} \right) + \sqrt {{x^2} + 3x} } \right| + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {{x^2} + 3x} $ is $\dfrac{{\left( {2x + 3} \right)}}{2}\sqrt {{x^2} + 3x}  - \dfrac{9}{8}\log \left| {\left( {x + \dfrac{3}{2}} \right) + \sqrt {{x^2} + 3x} } \right| + C$.

9. Integrate $\sqrt {1 + \dfrac{{{x^2}}}{9}} $.

Ans: Let $I = \int {\sqrt {1 + \dfrac{{{x^2}}}{9}} dx} $

$I = \dfrac{1}{3}\int {\sqrt {9 + {x^2}} dx} $

$I = \dfrac{1}{3}\int {\sqrt {{{\left( 3 \right)}^2} + {{\left( x \right)}^2}} dx} $

It is known that,

$\sqrt {{x^2} + {a^2}} dx = \dfrac{x}{2}\sqrt {{x^2} + {a^2}}  + \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} + {a^2}} } \right| + C$ 

For $a$ substitute $3$ and solve the integral.

$\therefore I = \dfrac{1}{3}\left[ {\dfrac{x}{2}\sqrt {{x^2} + 9}  + \dfrac{9}{2}\log \left| {x + \sqrt {{x^2} + 9} } \right|} \right] + C$

$ = \dfrac{x}{6}\sqrt {{x^2} + 9}  + \dfrac{3}{2}\log \left| {x + \sqrt {{x^2} + 9} } \right| + C$

Where $C$ is an arbitrary constant.

The integral of $\sqrt {1 + \dfrac{{{x^2}}}{9}} $ is $\dfrac{x}{6}\sqrt {{x^2} + 9}  + \dfrac{3}{2}\log \left| {x + \sqrt {{x^2} + 9} } \right| + C$.

10. $\int {\sqrt {1 + {x^2}} } $ Is equal to

A. $\dfrac{x}{2}\sqrt {1 + {x^2}}  + \dfrac{1}{2}\log \left| {x + \sqrt {1 + {x^2}} } \right| + C$

B. $\dfrac{2}{3}{\left( {1 + {x^2}} \right)^{\dfrac{2}{3}}} + C$ 

C. $\dfrac{2}{3}x{\left( {1 + {x^2}} \right)^{\dfrac{2}{3}}} + C$

D. $\dfrac{{{x^3}}}{2}\sqrt {1 + {x^2}}  + \dfrac{1}{2}{x^2}\log \left| {x + \sqrt {1 + {x^2}} } \right| + C$ 

Ans: It is known that,

$\sqrt {{x^2} + {a^2}} dx = \dfrac{x}{2}\sqrt {{x^2} + {a^2}}  + \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} + {a^2}} } \right| + C$

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

For $a$ substitute $1$ and solve the integral.

$\therefore \int {\sqrt {1 + {x^2}} } dx = \dfrac{x}{2}\sqrt {1 + {x^2}}  + \dfrac{1}{2}\log \left| {x + \sqrt {1 + {x^2}} } \right| + C$

Hence, the correct answer is A.

11. $\int {\sqrt {{x^2} - 8x + 7} } $ Is equal to

A. $\dfrac{1}{2}\left( {x - 4} \right)\sqrt {{x^2} - 8x + 7}  + 9\log \left| {x - 4 + \sqrt {{x^2} - 8x + 7} } \right| + C$

B. $\dfrac{1}{2}\left( {x + 4} \right)\sqrt {{x^2} - 8x + 7}  + 9\log \left| {x + 4 + \sqrt {{x^2} - 8x + 7} } \right| + C$ 

C. $\dfrac{1}{2}\left( {x - 4} \right)\sqrt {{x^2} - 8x + 7}  - 3\sqrt 2 \log \left| {x - 4 + \sqrt {{x^2} - 8x + 7} } \right| + C$

D. $\dfrac{1}{2}\left( {x - 4} \right)\sqrt {{x^2} - 8x + 7}  - \dfrac{9}{2}\log \left| {x - 4 + \sqrt {{x^2} - 8x + 7} } \right| + C$ 

Ans: Let $I = \int {\sqrt {{x^2} - 8x + 7} } dx$

$I = \int {\sqrt {\left( {{x^2} - 8x + 16} \right) - 9} dx} $

$I = \int {\sqrt {{{\left( {x - 4} \right)}^2} - {{\left( 3 \right)}^2}} dx} $

It is known that,

$\sqrt {{x^2} - {a^2}} dx = \dfrac{x}{2}\sqrt {{x^2} - {a^2}}  - \dfrac{{{a^2}}}{2}\log \left| {x + \sqrt {{x^2} - {a^2}} } \right| + C$ 

For $a$ substitute $3$ and for $x$ put $x - 4$ and solve the integral.

$\therefore I = \dfrac{{\left( {x - 4} \right)}}{2}\sqrt {{x^2} - 8x + 7}  - \dfrac{9}{2}\log \left| {\left( {x - 4} \right) + \sqrt {{x^2} - 8x + 7} } \right| + C$

Hence, the correct answer is D.

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.7

Opting for the NCERT solutions for Ex 7.7 Class 12 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 7.7 Class 12 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

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Besides these NCERT solutions for Class 12 Maths Chapter 7 Exercise 7.7, there are plenty of exercises in this chapter which contain innumerable questions as well. All these questions are solved/answered by our in-house subject experts as mentioned earlier. Hence all of these are bound to be of superior quality and anyone can refer to these during the time of exam preparation. In order to score the best possible marks in the class, it is really important to understand all the concepts of the textbooks and solve the problems from the exercises given next to it. 

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