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Q11:
Choose the correct answer in Exercises 10 to 11.
11. $\int \sqrt{x^2-8x+7} dx$ is equal to
Solution :
Step 1: Completing the Square
To integrate functions of the form $\sqrt{x^2 + bx + c}$, we must first complete the square for the quadratic expression $x^2 - 8x + 7$.
The expression is $x^2 - 8x + 7$. To complete the square, we take the coefficient of $x$ (which is $-8$), divide by $2$ to get $-4$, and square it to get $16$. We then add and subtract $16$:
$x^2 - 8x + 7 = (x^2 - 8x + 16) - 16 + 7$ [Algebraic identity: $(x-a)^2 = x^2 - 2ax + a^2$]
$= (x - 4)^2 - 9$
$= (x - 4)^2 - 3^2$
Step 2: Selecting the Integration Formula
The integral now takes the standard form $\int \sqrt{x^2 - a^2} \, dx$. The standard formula for this integral is:
$\int \sqrt{x^2 - a^2} \, dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2} \ln \left| x + \sqrt{x^2 - a^2} \right| + C$ [Standard Integral Formula]
Step 3: Substitution and Evaluation
Substitute $x$ with $(x - 4)$ and $a$ with $3$ into the standard formula:
$\int \sqrt{(x - 4)^2 - 3^2} \, dx = \frac{x - 4}{2}\sqrt{(x - 4)^2 - 3^2} - \frac{3^2}{2} \ln \left| (x - 4) + \sqrt{(x - 4)^2 - 3^2} \right| + C$
Simplify the terms:
- The term $\sqrt{(x - 4)^2 - 3^2}$ is equivalent to $\sqrt{x^2 - 8x + 7}$.
- The constant $\frac{3^2}{2} = \frac{9}{2}$.
Substituting these back into the expression:
$= \frac{x - 4}{2}\sqrt{x^2 - 8x + 7} - \frac{9}{2} \ln \left| (x - 4) + \sqrt{x^2 - 8x + 7} \right| + C$
Step 4: Visualizing the Function
Final Solution: $\frac{x-4}{2}\sqrt{x^2-8x+7} - \frac{9}{2} \ln |x-4 + \sqrt{x^2-8x+7}| + C$
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. $\sqrt{x^2+4x-5}$
- 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
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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