NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.7) Exercise 7.7
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Mathematics has often been upheld as an imperative subject necessary for the smooth and unhindered functioning of human life. Additionally, mathematics and its subdisciplines possess a considerable scope for academic research, which also implies that it is a field with many profitable career choices and promising opportunities. Mathematics is an indispensable academic discipline that is taught to students from the earliest stages of their schooling. Accordingly, it plays a considerable role in the prescribed NCERT academic curriculum for Class 12. Consequently, this necessitates the availability of and convenient access to reliable NCERT Solutions to aid selflearning. The NCERT curriculum for Class 12 Mathematics comprises thirteen distinct chapters that have been carefully compiled into an ordered sequence. Chapter 7 is titled Integrals. Of the entire prescribed curriculum for Class 12 by NCERT, integrals is one of the most versatile and complex themes with many underlying subtopics. Exercise 7.7, within the numerous challenging assessments that make up Chapter 7, is an important element that cannot be overlooked to excel in examinations.
The Extramarks’ website, in collaboration with renowned subject experts, has prepared the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 to enhance the conceptual clarity of students.
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NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex 7.7) Exercise 7.7
Students and teachers both have their share of concerns concerning the theme of integrals and calculus. The majority of students dislike the theorem of calculus because they consider it to be a major hindrance to academic excellence. However, students must take note of the fact that these themes are prominent and very crucial portions of the mathematics syllabus prescribed for Class 12. These are also very fundamental disciplines that possess vast potential for academic and scientific research. Therefore, their importance cannot be overlooked. It could prove to be challenging for students to follow the instruction and guidance of their teachers during classroom teaching, or to maintain their pace of grasping teachings in accordance with that of their peers. In such a situation, the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 might play a crucial role in encouraging selflearning among students. Extramarks has gone the extra mile in its efforts at compiling the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7, through intensive research and careful analysis to ensure the quality of the content. In the academic field of mathematics, integrals are an important area of research. This is why it is vital to lay a strong and reliable foundation by developing an adequate amount of conceptual clarity to prepare students appropriately for their future studies. The NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7, are detailed, highquality, and easytounderstand solutions to the questions covered in Exercise 7.7. The NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 would also help students seek appropriate answers to their doubts and queries while practising.
Extramarks is committed to providing dependable, genuine, and selfexplanatory NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 to motivate students to engage with the themes of integrals and calculus by moving away from their prior apprehensions. The wellorganized and logicallystructured format of the calculations provided as a part of the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7, would help students adequately adhere to the ideal format for problemsolving and would clarify their doubts stepbystep.
Access NCERT Solutions for Class 12 Chapter 7 – Integrals
The academic curriculum for Class 12 prescribed by the NCERT consists of the theme of integrals, which is widely acknowledged as a complex and versatile topic. On the one hand, it is a conceptually dynamic theme, and on the other hand, it also acts as an opening into a wider expanse of research and academic study for those who are looking forward to pursuing mathematics in the future as a career choice. Remarkably, students must regularly indulge in practising comprehensive and detailed assessments through the aid of quality reference materials like the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 to attain perfection in Integrals and Calculus. Regular practice of the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 would undoubtedly also result in a considerable improvement in the students’ memory retention and problemsolving capabilities. Moreover, the problemsolving skills acquired by students would equip them with the necessary understanding required to adequately decode the problem at hand and develop an appropriate response to it. Students would be able to choose the right practical approach to solving a problem along with the appropriate formulae that which have to be applied to do the necessary calculations. The NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 have been planned to function efficiently as consumable learning resources during the examination period for students. Therefore, the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 have been compiled very selectively through the inclusion of the most efficient ways of solving longdrawn problems related to the Integrals and theorem of Calculus.
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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.7
The NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 have been compiled through tireless efforts by Extramarks to cater to the nuances of the updated NCERT academic curriculum. The NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 have been organised cautiously to aid the selflearning of students in a way that they can appropriately follow the logical sequence of calculations. It is fundamental for students to acquire and retain crucial formulas and their derivatives. It is also equally important for them to be able to practically apply these formulae for solving problems. This is a skill that can be attained through constant practice and revision. Therefore, Extramarks provides the NCERT Solutions For Class 12 Maths Chapter 7 Exercise 7.7 as comprehensive and curated reference material with quality content for the convenience of reference whenever required.
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Q.1
$\begin{array}{l}\begin{array}{l}\mathrm{Choose}\text{}\mathrm{the}\text{}\mathrm{correct}\text{}\mathrm{answer}\\ \int \sqrt{1+{\text{x}}^{2}}\text{}\mathrm{dx}\mathrm{is}\mathrm{equal}\mathrm{to}\end{array}\\ \left(\text{A}\right)\text{}\frac{\text{x}}{2}\sqrt{1+{\text{x}}^{2}}+\frac{1}{2}\mathrm{log}\left\left(\text{x}+\sqrt{1+{\text{x}}^{2}}\right)\right+\text{C}\\ \left(\text{B}\right)\text{}\frac{2}{3}{\left(1+{\text{x}}^{2}\right)}^{\frac{3}{2}}+\text{C}\\ \left(\text{C}\right)\text{}\frac{2}{3}\text{x}{\left(1+{\text{x}}^{2}\right)}^{\frac{3}{2}}+\text{C}\\ \left(\text{D}\right)\text{}\frac{{\text{x}}^{2}}{2}\sqrt{1+{\text{x}}^{2}}\text{}+\frac{1}{2}{\text{x}}^{2}\text{}\mathrm{log}\left\left(\text{x}+\sqrt{1+{\text{x}}^{2}}\right)\right+\text{C}\end{array}$
Ans
$\begin{array}{l}\mathrm{Let}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{I}=\int \sqrt{1+{\mathrm{x}}^{2}}\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}}=\int \sqrt{{\mathrm{x}}^{2}+{1}^{2}}\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}[\mu \int \sqrt{{\mathrm{x}}^{2}+{\mathrm{a}}^{2}}\text{\hspace{0.17em}}\mathrm{dx}=\frac{\mathrm{x}}{2}\sqrt{{\mathrm{x}}^{2}+{\mathrm{a}}^{2}}+\frac{{\mathrm{a}}^{2}}{2}\mathrm{log}\mathrm{x}+\sqrt{{\mathrm{x}}^{2}+{\mathrm{a}}^{2}}]\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=(\frac{\mathrm{x}}{2}\sqrt{{\mathrm{x}}^{2}+{1}^{2}}+\frac{{1}^{2}}{2}\mathrm{log}\mathrm{x}+\sqrt{{\mathrm{x}}^{2}+{1}^{2}})+\mathrm{C}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}}=\frac{\mathrm{x}}{2}\sqrt{{\mathrm{x}}^{2}+1}+\frac{1}{2}\mathrm{log}\mathrm{x}+\sqrt{{\mathrm{x}}^{2}+1}+\mathrm{C}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{\mathrm{x}}{2}\sqrt{1+{\mathrm{x}}^{2}}+\frac{1}{2}\mathrm{log}\mathrm{x}+\sqrt{1+{\mathrm{x}}^{2}}+\mathrm{C}\\ \mathrm{Hence},\text{\xe2\u20ac\u2039\hspace{0.17em}\hspace{0.17em}}\mathrm{the}\text{correct option is A.}\end{array}$
Q.2
$\begin{array}{l}\begin{array}{l}\mathrm{Choose}\text{}\mathrm{the}\text{}\mathrm{correct}\text{}\mathrm{answer}\\ \int \sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}\text{}\mathrm{dx}\mathrm{is}\mathrm{equal}\mathrm{to}\end{array}\\ \left(\mathrm{A}\right)\text{}\frac{1}{2}\left(\mathrm{x}4\right)\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}+9\text{}\mathrm{log}\left\mathrm{x}4+\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}\right+\mathrm{C}\\ \left(\mathrm{B}\right)\text{}\frac{1}{4}\left(\mathrm{x}+4\right)\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}+9\mathrm{log}\left\mathrm{x}+4+\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}\right+\mathrm{C}\\ \left(\mathrm{C}\right)\text{}\frac{1}{2}\left(\mathrm{x}4\right)\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}3\sqrt{2}\text{}\mathrm{log}\left\mathrm{x}4+\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}\right+\mathrm{C}\\ \left(\mathrm{D}\right)\text{}\frac{1}{2}\left(\mathrm{x}4\right)\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}\frac{9}{2}\mathrm{log}\left\mathrm{x}4+\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}\right+\mathrm{C}\end{array}$
Ans
$\begin{array}{l}\mathrm{We}\text{have,}\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}=\sqrt{{(\mathrm{x}4)}^{2}9}\\ \mathrm{Let}\text{I}=\int \sqrt{{(\mathrm{x}4)}^{2}{\left(3\right)}^{2}}\text{\hspace{0.17em}}\mathrm{dx}\\ \text{\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\int \sqrt{{\mathrm{t}}^{2}{\left(3\right)}^{2}}\text{\hspace{0.17em}}\mathrm{dt}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left[\mathrm{Let}\text{\hspace{0.17em}t}=\mathrm{x}4\Rightarrow \frac{\mathrm{dt}}{\mathrm{dx}}=1\right]\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}[\mu \int \sqrt{{\mathrm{x}}^{2}{\mathrm{a}}^{2}}\text{\hspace{0.17em}}\mathrm{dx}=\frac{\mathrm{x}}{2}\sqrt{{\mathrm{x}}^{2}{\mathrm{a}}^{2}}\frac{{\mathrm{a}}^{2}}{2}\mathrm{log}\mathrm{x}+\sqrt{{\mathrm{x}}^{2}{\mathrm{a}}^{2}}]\\ \text{\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{\mathrm{t}}{2}\sqrt{{\mathrm{t}}^{2}{\left(3\right)}^{2}}\frac{{\left(3\right)}^{2}}{2}\mathrm{log}\mathrm{t}+\sqrt{{\mathrm{t}}^{2}{\left(3\right)}^{2}}+\mathrm{C}\\ \mathrm{Putting}\text{t}=\mathrm{x}4,\text{\hspace{0.17em}}\mathrm{we}\text{\hspace{0.17em}}\mathrm{get}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{(\mathrm{x}4)}{2}\sqrt{{(\mathrm{x}4)}^{2}{\left(3\right)}^{2}}\frac{9}{2}\mathrm{log}(\mathrm{x}4)+\sqrt{{(\mathrm{x}4)}^{2}{\left(3\right)}^{2}}+\mathrm{C}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\frac{(\mathrm{x}4)}{2}\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}\frac{9}{2}\mathrm{log}(\mathrm{x}4)+\sqrt{{\mathrm{x}}^{2}8\mathrm{x}+7}+\mathrm{C}\\ \mathrm{Hence},\text{the correct option is D.}\end{array}$
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