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Q5:
By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.
5. $\int_{-5}^{5} |x+2| dx$
Solution :
Step 1: Identifying the Property of Absolute Value
Given the integral $I = \int_{-5}^{5} |x+2| \, dx$. The integrand is an absolute value function defined as:
$|x+2| = \begin{cases} -(x+2) & \text{if } x < -2 \\ (x+2) & \text{if } x \ge -2 \end{cases}$
[By the definition of the absolute value function $|f(x)|$]
Step 2: Splitting the Interval of Integration
Since the integrand changes its definition at $x = -2$, we split the integral at this point within the given range $[-5, 5]$:
$I = \int_{-5}^{-2} -(x+2) \, dx + \int_{-2}^{5} (x+2) \, dx$
[By the Additivity Property of Definite Integrals: $\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx$ where $a < b < c$]
Step 3: Evaluating the First Integral ($I_1$)
$I_1 = \int_{-5}^{-2} -(x+2) \, dx = - \left[ \frac{x^2}{2} + 2x \right]_{-5}^{-2}$
$I_1 = - \left[ \left( \frac{(-2)^2}{2} + 2(-2) \right) - \left( \frac{(-5)^2}{2} + 2(-5) \right) \right]$
$I_1 = - \left[ (2 - 4) - (\frac{25}{2} - 10) \right] = - \left[ -2 - (\frac{5}{2}) \right] = - \left[ -\frac{9}{2} \right] = \frac{9}{2}$
Step 4: Evaluating the Second Integral ($I_2$)
$I_2 = \int_{-2}^{5} (x+2) \, dx = \left[ \frac{x^2}{2} + 2x \right]_{-2}^{5}$
$I_2 = \left[ \left( \frac{5^2}{2} + 2(5) \right) - \left( \frac{(-2)^2}{2} + 2(-2) \right) \right]$
$I_2 = \left[ (\frac{25}{2} + 10) - (2 - 4) \right] = \left[ \frac{45}{2} - (-2) \right] = \frac{45}{2} + 2 = \frac{49}{2}$
Step 5: Visual Representation of the Area
Step 6: Summing the Components
$I = I_1 + I_2$
$I = \frac{9}{2} + \frac{49}{2} = \frac{58}{2} = 29$
Final Solution: 29
More Questions from Class 12 Mathematics Integrals EXERCISE 7.10
- Q1: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 1. $\int_{0}^{\frac{\pi}{2}} \cos^2 x dx$
- Q2: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 2. $\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$
- Q3: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 3. $\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{3}{2}} x dx}{\sin^{\frac{3}{2}} x + \cos^{\frac{3}{2}} x}$
- Q4: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 4. $\int_{0}^{\frac{\pi}{2}} \frac{\cos^5 x dx}{\sin^5 x + \cos^5 x}$
- Q6: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 6. $\int_{2}^{8} |x-5| dx$
- Q7: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 7. $\int_{0}^{1} x(1-x)^n dx$
- Q8: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 8. $\int_{0}^{\frac{\pi}{4}} \log(1+\tan x) dx$
- Q9: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 9. $\int_{0}^{2} x \sqrt{2-x} dx$
- Q10: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 10. $\int_{0}^{\frac{\pi}{2}} (2\log \sin x - \log \sin 2x) dx$
- Q11: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 11. $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^2 x dx$
- Q12: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 12. $\int_{0}^{\pi} \frac{x dx}{1+\sin x}$
- Q13: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 13. $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x dx$
- Q14: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 14. $\int_{0}^{2\pi} \cos^5 x dx$
- Q15: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 15. $\int_{0}^{\frac{\pi}{2}} \frac{\sin x - \cos x}{1+\sin x \cos x} dx$
- Q16: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 16. $\int_{0}^{\pi} \log(1+\cos x) dx$
- Q17: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 17. $\int_{0}^{a} \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a-x}} dx$
- Q18: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 18. $\int_{0}^{4} |x-1| dx$
- Q19: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 19. Show that $\int_{0}^{a} f(x) g(x) dx = 2 \int_{0}^{a} f(x) dx$, if $f$ and $g$ are defined as $f(x) = f(a-x)$ and $g(x) + g(a-x) = 4$
- Q20: Choose the correct answer in Exercises 20 and 21. 20. The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x^3 + x \cos x + \tan^5 x + 1) dx$ is
- Q21: Choose the correct answer in Exercises 20 and 21. 21. The value of $\int_{0}^{\frac{\pi}{2}} \log (\frac{4+3\sin x}{4+3\cos x}) dx$ is
CBSE Solutions for Class 12 Mathematics Integrals
- Integrals Class 12 EXERCISE 7.1
- 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 7.7
- 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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