- Ex. 7.1
- Ex. 7.2
- Ex. 7.2
- Ex. 7.3
- Ex. 7.4
- Ex. 7.5
- Ex. 7.6
- Ex. 7.7
- Ex. 7.8
- Ex. 7.9
- Ex. 7.10
- Ex. 7.11
- Miscellaneous Exercise
Integrals Class 12th Mathematics Part II CBSE Solution - Exercise 7.7
This exercise deals with the integration of some standard functions involving square roots of quadratic expressions. The key formulas used in this exercise are:
- \(\int \sqrt{x^2 - a^2} dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\log|x + \sqrt{x^2 - a^2}| + C\)
- \(\int \sqrt{x^2 + a^2} dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log|x + \sqrt{x^2 + a^2}| + C\)
- \(\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C\)
Solution:
Let \( I = \int \sqrt{4-x^2} dx \)
This can be written as:
\( I = \int \sqrt{(2)^2 - x^2} dx \)
We know that \(\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C\)
Here, \(a = 2\). Substituting the values:
\( I = \frac{x}{2}\sqrt{4-x^2} + \frac{2^2}{2}\sin^{-1}\frac{x}{2} + C \)
\( \Rightarrow I = \frac{x}{2}\sqrt{4-x^2} + 2\sin^{-1}\frac{x}{2} + C \)
Solution:
Let \( I = \int \sqrt{1-4x^2} dx = \int \sqrt{1-(2x)^2} dx \)
Let \( 2x = t \), then \( 2dx = dt \Rightarrow dx = \frac{dt}{2} \)
\( I = \frac{1}{2}\int \sqrt{1^2 - t^2} dt \)
Using the formula \(\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C\):
\( I = \frac{1}{2}\left[ \frac{t}{2}\sqrt{1-t^2} + \frac{1}{2}\sin^{-1} t \right] + C \)
Substitute \( t = 2x \):
\( I = \frac{1}{2}\left[ \frac{2x}{2}\sqrt{1-4x^2} + \frac{1}{2}\sin^{-1}(2x) \right] + C \)
\( \Rightarrow I = \frac{x}{2}\sqrt{1-4x^2} + \frac{1}{4}\sin^{-1}(2x) + C \)
Solution:
Let \( I = \int \sqrt{x^2 + 4x + 6} dx \)
First, complete the square inside the root:
\( x^2 + 4x + 6 = x^2 + 4x + 4 + 2 = (x+2)^2 + (\sqrt{2})^2 \)
\( I = \int \sqrt{(x+2)^2 + (\sqrt{2})^2} dx \)
Using the formula \(\int \sqrt{X^2 + a^2} dx = \frac{X}{2}\sqrt{X^2 + a^2} + \frac{a^2}{2}\log|X + \sqrt{X^2 + a^2}| + C\):
Here \( X = x+2 \) and \( a = \sqrt{2} \).
\( I = \frac{x+2}{2}\sqrt{x^2 + 4x + 6} + \frac{2}{2}\log| (x+2) + \sqrt{x^2 + 4x + 6} | + C \)
\( \Rightarrow I = \frac{x+2}{2}\sqrt{x^2 + 4x + 6} + \log| x+2 + \sqrt{x^2 + 4x + 6} | + C \)
Solution:
Let \( I = \int \sqrt{x^2 + 4x + 1} dx \)
Completing the square:
\( x^2 + 4x + 1 = x^2 + 4x + 4 - 3 = (x+2)^2 - (\sqrt{3})^2 \)
\( I = \int \sqrt{(x+2)^2 - (\sqrt{3})^2} dx \)
Using the formula \(\int \sqrt{X^2 - a^2} dx = \frac{X}{2}\sqrt{X^2 - a^2} - \frac{a^2}{2}\log|X + \sqrt{X^2 - a^2}| + C\):
\( I = \frac{x+2}{2}\sqrt{x^2 + 4x + 1} - \frac{3}{2}\log| (x+2) + \sqrt{x^2 + 4x + 1} | + C \)
Solution:
Let \( I = \int \sqrt{1 - 4x - x^2} dx \)
Rearranging the term inside the root:
\( 1 - (x^2 + 4x) = 1 - (x^2 + 4x + 4 - 4) = 1 - ((x+2)^2 - 4) = 5 - (x+2)^2 \)
\( I = \int \sqrt{(\sqrt{5})^2 - (x+2)^2} dx \)
Using the formula \(\int \sqrt{a^2 - X^2} dx = \frac{X}{2}\sqrt{a^2 - X^2} + \frac{a^2}{2}\sin^{-1}\frac{X}{a} + C\):
\( I = \frac{x+2}{2}\sqrt{1 - 4x - x^2} + \frac{5}{2}\sin^{-1}\left( \frac{x+2}{\sqrt{5}} \right) + C \)
Solution:
Let \( I = \int \sqrt{x^2 + 4x - 5} dx \)
Completing the square:
\( x^2 + 4x - 5 = x^2 + 4x + 4 - 9 = (x+2)^2 - 3^2 \)
\( I = \int \sqrt{(x+2)^2 - 3^2} dx \)
Using the formula \(\int \sqrt{X^2 - a^2} dx = \frac{X}{2}\sqrt{X^2 - a^2} - \frac{a^2}{2}\log|X + \sqrt{X^2 - a^2}| + C\):
\( I = \frac{x+2}{2}\sqrt{x^2 + 4x - 5} - \frac{9}{2}\log| (x+2) + \sqrt{x^2 + 4x - 5} | + C \)
Solution:
Let \( I = \int \sqrt{1 + 3x - x^2} dx \)
Completing the square for \( -(x^2 - 3x - 1) \):
\( 1 - (x^2 - 3x) = 1 - (x^2 - 3x + \frac{9}{4} - \frac{9}{4}) \)
\( = 1 - (x - \frac{3}{2})^2 + \frac{9}{4} = \frac{13}{4} - (x - \frac{3}{2})^2 \)
\( I = \int \sqrt{ \left(\frac{\sqrt{13}}{2}\right)^2 - \left(x - \frac{3}{2}\right)^2 } dx \)
Using \(\int \sqrt{a^2 - X^2} dx\):
\( I = \frac{x - \frac{3}{2}}{2}\sqrt{1 + 3x - x^2} + \frac{\frac{13}{4}}{2}\sin^{-1}\left( \frac{x - \frac{3}{2}}{\frac{\sqrt{13}}{2}} \right) + C \)
Simplifying:
\( I = \frac{2x - 3}{4}\sqrt{1 + 3x - x^2} + \frac{13}{8}\sin^{-1}\left( \frac{2x - 3}{\sqrt{13}} \right) + C \)
Solution:
Let \( I = \int \sqrt{x^2 + 3x} dx \)
Completing the square:
\( x^2 + 3x = x^2 + 3x + \frac{9}{4} - \frac{9}{4} = (x + \frac{3}{2})^2 - (\frac{3}{2})^2 \)
\( I = \int \sqrt{ (x + \frac{3}{2})^2 - (\frac{3}{2})^2 } dx \)
Using \(\int \sqrt{X^2 - a^2} dx\):
\( I = \frac{x + \frac{3}{2}}{2}\sqrt{x^2 + 3x} - \frac{\frac{9}{4}}{2}\log| (x + \frac{3}{2}) + \sqrt{x^2 + 3x} | + C \)
\( \Rightarrow I = \frac{2x + 3}{4}\sqrt{x^2 + 3x} - \frac{9}{8}\log| x + \frac{3}{2} + \sqrt{x^2 + 3x} | + C \)
Solution:
Let \( I = \int \sqrt{1 + \frac{x^2}{9}} dx = \int \sqrt{\frac{9 + x^2}{9}} dx \)
\( I = \frac{1}{3}\int \sqrt{x^2 + 3^2} dx \)
Using \(\int \sqrt{X^2 + a^2} dx\):
\( I = \frac{1}{3}\left[ \frac{x}{2}\sqrt{x^2 + 9} + \frac{9}{2}\log| x + \sqrt{x^2 + 9} | \right] + C \)
\( \Rightarrow I = \frac{x}{6}\sqrt{x^2 + 9} + \frac{3}{2}\log| x + \sqrt{x^2 + 9} | + C \)
A. \(\frac{x}{2}\sqrt{1+x^2} + \frac{1}{2}\log|x+\sqrt{1+x^2}| + C\)
B. \(\frac{2}{3}(1+x^2)^{3/2} + C\)
C. \(\frac{2}{3}x(1+x^2)^{3/2} + C\)
D. \(\frac{x}{2}\sqrt{1+x^2} + \frac{1}{2}x^2\log|x+\sqrt{1+x^2}| + C\)
Correct Answer: A
Explanation:
We know that \(\int \sqrt{x^2 + a^2} dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log|x + \sqrt{x^2 + a^2}| + C\).
Here \( a = 1 \).
Therefore, \(\int \sqrt{1+x^2} dx = \frac{x}{2}\sqrt{1+x^2} + \frac{1}{2}\log|x+\sqrt{1+x^2}| + C\).
A. \(\frac{1}{2}(x-4)\sqrt{x^2 - 8x + 7} + 9\log|x - 4 + \sqrt{x^2 - 8x + 7}| + C\)
B. \(\frac{1}{2}(x+4)\sqrt{x^2 - 8x + 7} + 9\log|x + 4 + \sqrt{x^2 - 8x + 7}| + C\)
C. \(\frac{1}{2}(x-4)\sqrt{x^2 - 8x + 7} - 3\sqrt{2}\log|x - 4 + \sqrt{x^2 - 8x + 7}| + C\)
D. \(\frac{1}{2}(x-4)\sqrt{x^2 - 8x + 7} - \frac{9}{2}\log|x - 4 + \sqrt{x^2 - 8x + 7}| + C\)
Correct Answer: D
Explanation:
Let \( I = \int \sqrt{x^2 - 8x + 7} dx \)
Completing the square: \( x^2 - 8x + 7 = (x-4)^2 - 16 + 7 = (x-4)^2 - 9 = (x-4)^2 - 3^2 \)
\( I = \int \sqrt{(x-4)^2 - 3^2} dx \)
Using the formula: \(\int \sqrt{X^2 - a^2} dx = \frac{X}{2}\sqrt{X^2 - a^2} - \frac{a^2}{2}\log|X + \sqrt{X^2 - a^2}| + C\)
Here \( X = x-4 \) and \( a = 3 \).
\( I = \frac{x-4}{2}\sqrt{x^2 - 8x + 7} - \frac{9}{2}\log| (x-4) + \sqrt{x^2 - 8x + 7} | + C \)
This matches option D.