These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7 Questions and Answers are prepared by our highly skilled subject experts. https://mcq-questions.com/ncert-solutions-for-class-12-maths-chapter-7-ex-7-7/
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.7
Question 1.
\(\sqrt{4-x^{2}}\)
Solution:
\(\sqrt{4-x^{2}}\) dx
= \(\int \sqrt{2^{2}-x^{2}} d x\)
= \(\frac{x}{2} \sqrt{2^{2}-x^{2}}+\frac{4}{2} \sin ^{-1} \frac{x}{2}+C\)
= \(\frac{x}{2} \sqrt{4-x^{2}}+2 \sin ^{-1} \frac{x}{2}+C\)
Question 2.
\(\sqrt { 1-{ 4x }^{ 2 } } \)
Solution:
Question 3.
\(\sqrt { { x }^{ 2 }+4x+6 } \)
Solution:
\(\sqrt { { x }^{ 2 }+4x+6 } \) dx
= \(\int \sqrt{x^{2}+4 x+4+6-4} d x=\int \sqrt{(x+2)^{2}+(\sqrt{2})^{2}} d x\)
= \(\frac{x+2}{2} \sqrt{(x+2)^{2}+(\sqrt{2})^{2}}+\frac{2}{2} \log \left|(x+2)+\sqrt{(x+2)^{2}+(\sqrt{2})^{2}}\right|+\mathrm{C}\)
= \(\frac{(x+2)}{2} \sqrt{x^{2}+4 x+6}+\log \left|(x+2)+\sqrt{x^{2}+4 x+6}\right|+C\)
Question 4.
\(\sqrt{x^{2}+4 x+1}\)
Solution:
\(\sqrt{x^{2}+4 x+1}\) dx
= \(\int \sqrt{x^{2}+4 x+4+1-4} d x=\int \sqrt{(x+2)^{2}-(\sqrt{3})^{2}} d x\)
= \(\frac{(x+2)}{2} \sqrt{(x+2)^{2}-(\sqrt{3})^{2}}-\frac{3}{2} \log \left|(x+2)+\sqrt{(x+2)^{2}-(\sqrt{3})^{2}}\right|+\mathrm{C}\)
= \(\frac{(x+2)}{2} \sqrt{x^{2}+4 x+1}-\frac{3}{2} \log \left|(x+2) \sqrt{x^{2}+4 x+1}\right|+C\)
Question 5.
\(\sqrt { 1-4x-{ x }^{ 2 } }\)
Solution:
Question 6.
\(\sqrt { { x }^{ 2 }+4x-5 } \)
Solution:
\(\sqrt { { x }^{ 2 }+4x-5 } \)dx
\(\int { \sqrt { { x }^{ 2 }+4x-5 } }\) dx = \(\int { \sqrt { { (x+2) }^{ 2 }-{ (3) }^{ 2 } } }\)dx
= \(\frac { x+2 }{ 2 } \sqrt { { x }^{ 2 }+4x-5 } -\frac { 9 }{ 2 } log|x+2+\sqrt { { x }^{ 2 }+4x-5 } |+c\)
Question 7.
\(\sqrt { 1+3x-{ x }^{ 2 } } \)
Solution:
Question 8.
\(\sqrt { { x }^{ 2 }+3x } \)
Solution:
Question 9.
\(\sqrt { 1+\frac { { x }^{ 2 } }{ 9 } } \)
Solution:
Question 10.
\(\int \sqrt{1+x^{2}}\) dx is equal 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) \(\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log|x+\sqrt { 1+{ x }^{ 2 } } |+C\)
\(\int { \sqrt { 1+{ x }^{ 2 } } } \)dx
= \(\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log|x+\sqrt { 1+{ x }^{ 2 } } |+C\)
Question 11.
\(\int \sqrt{x^{2}-8 x+7}\) dx
Solution:
Question 12.
Integrate \(x \sqrt{x+x^{2}}\) w.r.t. x.
Solution:
I = \(x \sqrt{x+x^{2}}\) dx
Let x = A\(\frac { d }{ dx }\)(x + x²) + Bx = A(1 + 2x) + B
Equating coefficients of x both sides, we get 2A = 1 ⇒ A = \(\frac { 1 }{ 2 }\)
Equating constants on both sides, we get A + B = 0 ⇒ B = \(\frac { – 1 }{ 2 }\)
Question 13.
Integrate \((x+1) \sqrt{2 x^{2}+3}\) w.r.t. x.
Solution:
I = \((x+1) \sqrt{2 x^{2}+3}\) dx
Let x + 1 = A\(\frac { d }{ dx }\)(2x² + 3) + Bx = A(4x) + B
Equating the coefficients of x both sides, we get 4A = 1 ⇒ A = \(\frac { 1 }{ 4 }\)
Equating constants on both sides, we get B = 1
x + 1 = \(\frac { 1 }{ 4 }\)(4x) + 1
Question 14.
Integrate \((x+3) \sqrt{3-4 x-x^{2}}\) w.r.t. x.
Solution:
I = \((x+3) \sqrt{3-4 x-x^{2}}\) dx
Let x + 3 = A\(\frac { d }{ dx }\)(3 – 4x – x²) + Bx + 3 = A(- 4 – 2x) + B
Equating the coefficients of x both sides, we get -2A = 1 ⇒ A = \(\frac { – 1 }{ 2 }\)
Equating constants on both sides, we get -4A + B = 3 ⇒ B = 1
∴ x + 3 = \(\frac { – 1 }{ 2 }\)(- 4 – 2x) + 1
