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Q7:
Integrate the functions in Exercises 1 to 9.
7. $\sqrt{1+3x-x^2}$
Solution :
Step 1: Identifying the Form
The integral to be solved is $I = \int \sqrt{1 + 3x - x^2} \, dx$. This expression is of the form $\int \sqrt{a^2 - x^2} \, dx$, where the quadratic expression under the radical requires completion of the square [Completing the Square Method].
Step 2: Completing the Square
Rewrite the expression $1 + 3x - x^2$:
$1 + 3x - x^2 = -(x^2 - 3x - 1)$
To complete the square for $x^2 - 3x$, we add and subtract $(\frac{3}{2})^2 = \frac{9}{4}$:
$= -[x^2 - 3x + \frac{9}{4} - \frac{9}{4} - 1]$
$= -[(x - \frac{3}{2})^2 - \frac{13}{4}]$
$= \frac{13}{4} - (x - \frac{3}{2})^2$
Thus, the integral becomes: $I = \int \sqrt{(\frac{\sqrt{13}}{2})^2 - (x - \frac{3}{2})^2} \, dx$
Step 3: Applying the Standard Integral Formula
We utilize the standard result: [$\int \sqrt{a^2 - u^2} \, du = \frac{u}{2} \sqrt{a^2 - u^2} + \frac{a^2}{2} \sin^{-1}(\frac{u}{a}) + C$].
Here, $a = \frac{\sqrt{13}}{2}$ and $u = (x - \frac{3}{2})$.
$I = \frac{x - \frac{3}{2}}{2} \sqrt{\frac{13}{4} - (x - \frac{3}{2})^2} + \frac{\frac{13}{4}}{2} \sin^{-1}\left(\frac{x - \frac{3}{2}}{\frac{\sqrt{13}}{2}}\right) + C$
Step 4: Algebraic Simplification
Simplify the term $(x - \frac{3}{2})$ as $(\frac{2x - 3}{2})$:
$I = \frac{2x - 3}{4} \sqrt{1 + 3x - x^2} + \frac{13}{8} \sin^{-1}\left(\frac{2x - 3}{\sqrt{13}}\right) + C$
Step 5: Final Result
Combining the terms into the final analytical form:
Final Solution: $\frac{2x-3}{4}\sqrt{1+3x-x^2} + \frac{13}{8}\sin^{-1}\left(\frac{2x-3}{\sqrt{13}}\right) + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.7
- Q1: Integrate the functions in Exercises 1 to 9. 1. $\sqrt{4-x^2}$
- Q2: Integrate the functions in Exercises 1 to 9. 2. $\sqrt{1-4x^2}$
- Q3: Integrate the functions in Exercises 1 to 9. 3. $\sqrt{x^2+4x+6}$
- Q4: Integrate the functions in Exercises 1 to 9. 4. $\sqrt{x^2+4x+1}$
- Q5: Integrate the functions in Exercises 1 to 9. 5. $\sqrt{1-4x-x^2}$
- Q6: Integrate the functions in Exercises 1 to 9. 6. $\sqrt{x^2+4x-5}$
- Q8: Integrate the functions in Exercises 1 to 9. 8. $\sqrt{x^2+3x}$
- Q9: Integrate the functions in Exercises 1 to 9. 9. $\sqrt{1+\frac{x^2}{9}}$
- Q10: Choose the correct answer in Exercises 10 to 11. 10. $\int \sqrt{1+x^2} dx$ is equal to
- Q11: Choose the correct answer in Exercises 10 to 11. 11. $\int \sqrt{x^2-8x+7} dx$ is equal to
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- Integrals Class 12 EXERCISE 7.10
- Integrals Class 12 EXERCISE 7.2
- Integrals Class 12 EXERCISE 7.3
- Integrals Class 12 EXERCISE 7.4
- Integrals Class 12 EXERCISE 7.5
- Integrals Class 12 EXERCISE 7.6
- Integrals Class 12 EXERCISE 7.8
- Integrals Class 12 EXERCISE 7.9
- Integrals Class 12 Miscellaneous Exercise on Chapter 7
Chapters in CBSE - Class 12 Mathematics
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