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Q3:
Integrate the functions in Exercises 1 to 9.
3. $\sqrt{x^2+4x+6}$
Solution :
Step 1: Completing the Square for the Quadratic Form
The integral to evaluate is $I = \int \sqrt{x^2 + 4x + 6} \, dx$. To simplify the radicand, we complete the square for the quadratic expression $x^2 + 4x + 6$.
We write the expression as:
$x^2 + 4x + 6 = (x^2 + 4x + 4) - 4 + 6$[By the method of completing the square: $(x+a)^2 = x^2 + 2ax + a^2$ where $2a = 4 \implies a=2$]
$x^2 + 4x + 6 = (x+2)^2 + 2$ $x^2 + 4x + 6 = (x+2)^2 + (\sqrt{2})^2$Step 2: Selecting the Standard Integration Formula
The integral now takes the form $\int \sqrt{u^2 + a^2} \, du$, where $u = (x+2)$ and $a = \sqrt{2}$. The standard integration formula for this form is:
$\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2} \sqrt{x^2 + a^2} + \frac{a^2}{2} \ln|x + \sqrt{x^2 + a^2}| + C$
[Reference: Standard Integral Formula for $\sqrt{x^2 + a^2}$]
Step 3: Substitution and Integration
Applying the formula with $u = (x+2)$ and $a^2 = 2$:
$I = \int \sqrt{(x+2)^2 + (\sqrt{2})^2} \, dx$Following the formula:
$I = \frac{(x+2)}{2} \sqrt{(x+2)^2 + 2} + \frac{2}{2} \ln|(x+2) + \sqrt{(x+2)^2 + 2}| + C$Step 4: Simplification of the Result
Substitute $(x+2)^2 + 2$ back with the original expression $x^2 + 4x + 6$:
$I = \frac{x+2}{2} \sqrt{x^2 + 4x + 6} + 1 \cdot \ln|(x+2) + \sqrt{x^2 + 4x + 6}| + C$Final Solution: $\frac{(x+2)}{2} \sqrt{x^2 + 4x + 6} + \ln|x + 2 + \sqrt{x^2 + 4x + 6}| + C$
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. $\sqrt{1-4x^2}$
- 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. $\sqrt{x^2+4x-5}$
- Q7: Integrate the functions in Exercises 1 to 9. 7. $\sqrt{1+3x-x^2}$
- 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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