Exercise 7.7, Question 1
\(\displaystyle\int \sqrt{4 - x^2}\,dx\)
Show solution
- \(a = 2\) in the first form.
Answer: \[\dfrac{x}{2}\sqrt{4 - x^2} + 2\sin^{-1}\dfrac{x}{2} + C\]
NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.7
Exercise 7.7: Integrals of square roots of quadratics. \(\int\sqrt{a^2 - x^2}\,dx = \dfrac{x}{2}\sqrt{a^2 - x^2} + \dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}\), \(\int\sqrt{x^2 \pm a^2}\,dx = \dfrac{x}{2}\sqrt{x^2 \pm a^2} \pm \dfrac{a^2}{2}\log\left|x + \sqrt{x^2 \pm a^2}\right|\). Complete the square first and shift x.
- 11 questions
- Every answer checked by computer algebra
- Free, no sign-in
Exercise 7.7 questions and solutions
Exercise 7.7, Question 2
\(\displaystyle\int \sqrt{1 - 4x^2}\,dx\)
Show solution
- \(t = 2x\): \[\tfrac12\int\sqrt{1 - t^2}\,dt = \tfrac12\left[\dfrac{t}{2}\sqrt{1 - t^2} + \tfrac12\sin^{-1}t\right]\]
Answer: \[\dfrac{x}{2}\sqrt{1 - 4x^2} + \tfrac14\sin^{-1}2x + C\]
Exercise 7.7, Question 3
\(\displaystyle\int \sqrt{x^2 + 4x + 6}\,dx\)
Show solution
- \((x + 2)^2 + (\sqrt2)^2\): the \(x^2 + a^2\) form with \(a^2 = 2\).
Answer: \[\dfrac{x + 2}{2}\sqrt{x^2 + 4x + 6} + \log\left|x + 2 + \sqrt{x^2 + 4x + 6}\right| + C\]
Exercise 7.7, Question 4
\(\displaystyle\int \sqrt{x^2 + 4x + 1}\,dx\)
Show solution
- \((x + 2)^2 - (\sqrt3)^2\).
Answer: \[\dfrac{x + 2}{2}\sqrt{x^2 + 4x + 1} - \tfrac32\log\left|x + 2 + \sqrt{x^2 + 4x + 1}\right| + C\]
Exercise 7.7, Question 5
\(\displaystyle\int \sqrt{1 - 4x - x^2}\,dx\)
Show solution
- \(1 - 4x - x^2 = 5 - (x + 2)^2\).
Answer: \[\dfrac{x + 2}{2}\sqrt{1 - 4x - x^2} + \tfrac52\sin^{-1}\dfrac{x + 2}{\sqrt5} + C\]
Exercise 7.7, Question 6
\(\displaystyle\int \sqrt{x^2 + 4x - 5}\,dx\)
Show solution
- \((x + 2)^2 - 3^2\).
Answer: \[\dfrac{x + 2}{2}\sqrt{x^2 + 4x - 5} - \tfrac92\log\left|x + 2 + \sqrt{x^2 + 4x - 5}\right| + C\]
Exercise 7.7, Question 7
\(\displaystyle\int \sqrt{1 + 3x - x^2}\,dx\)
Show solution
- \[1 + 3x - x^2 = \tfrac{13}{4} - \left(x - \tfrac32\right)^2\]
Answer: \[\dfrac{2x - 3}{4}\sqrt{1 + 3x - x^2} + \dfrac{13}{8}\sin^{-1}\dfrac{2x - 3}{\sqrt{13}} + C\]
Exercise 7.7, Question 8
\(\displaystyle\int \sqrt{x^2 + 3x}\,dx\)
Show solution
- \[x^2 + 3x = \left(x + \tfrac32\right)^2 - \tfrac94\]
Answer: \[\dfrac{2x + 3}{4}\sqrt{x^2 + 3x} - \tfrac98\log\left|x + \tfrac32 + \sqrt{x^2 + 3x}\right| + C\]
Exercise 7.7, Question 9
\(\displaystyle\int \sqrt{1 + \dfrac{x^2}{9}}\,dx\)
Show solution
- \[\tfrac13\int\sqrt{x^2 + 9}\,dx = \tfrac13\left[\dfrac{x}{2}\sqrt{x^2 + 9} + \dfrac92\log\left|x + \sqrt{x^2 + 9}\right|\right]\]
Answer: \[\dfrac{x}{6}\sqrt{x^2 + 9} + \tfrac32\log\left|x + \sqrt{x^2 + 9}\right| + C\]
Exercise 7.7, Question 10
\(\displaystyle\int\sqrt{1 + x^2}\,dx\) equals: (A) \(\dfrac{x}{2}\sqrt{1 + x^2} + \tfrac12\log\left|x + \sqrt{1 + x^2}\right| + C\) (B) \(\tfrac23(1 + x^2)^{3/2} + C\) (C) \(\tfrac23x(1 + x^2)^{3/2} + C\) (D) \(\dfrac{x^2}{2}\sqrt{1 + x^2} + \tfrac12x^2\log\left|x + \sqrt{1 + x^2}\right| + C\)
Show solution
- The \(x^2 + a^2\) form with \(a = 1\).
Answer: (A)
Exercise 7.7, Question 11
\(\displaystyle\int\sqrt{x^2 - 8x + 7}\,dx\) equals: (A) \(\tfrac12(x - 4)\sqrt{x^2 - 8x + 7} + 9\log\left|x - 4 + \sqrt{x^2 - 8x + 7}\right| + C\) (B) \(\tfrac12(x + 4)\sqrt{x^2 - 8x + 7} + 9\log\left|x + 4 + \sqrt{x^2 - 8x + 7}\right| + C\) (C) \(\tfrac12(x - 4)\sqrt{x^2 - 8x + 7} - 3\sqrt2\log\left|x - 4 + \sqrt{x^2 - 8x + 7}\right| + C\) (D) \(\tfrac12(x - 4)\sqrt{x^2 - 8x + 7} - \tfrac92\log\left|x - 4 + \sqrt{x^2 - 8x + 7}\right| + C\)
Show solution
- \(x^2 - 8x + 7 = (x - 4)^2 - 3^2\).
Answer: (D)
Done the NCERT exercises? The board paper asks more
Integrals has 43 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Integrals in our sample papers: Sample paper 1 (questions 12, 23, 27, 29, 34) · Sample paper 2 (questions 13, 14, 24, 27) · Sample paper 3 (questions 11, 23, 28, 34) · Sample paper 4 (questions 13, 14, 24, 27, 37) · Sample paper 5 (questions 11, 23, 28, 29).
Also useful: free MCQs and case studies for Integrals · formulas for this chapter (Class 12 formula sheet, free PDF) · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.