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Q14:
By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.
14. $\int_{0}^{2\pi} \cos^5 x dx$
Solution :
Step 1: Application of Definite Integral Properties
We are tasked with evaluating the definite integral:
$I = \int_{0}^{2\pi} \cos^5 x \, dx$
To solve this, we apply the property of definite integrals involving symmetric bounds, specifically the property regarding periodicity: $\int_{0}^{2a} f(x) \, dx$ where $f(2a - x) = f(x)$. [Per the property: $\int_{0}^{2a} f(x) \, dx = 2\int_{0}^{a} f(x) \, dx$ if $f(2a-x) = f(x)$, and $0$ if $f(2a-x) = -f(x)$].
Step 2: Testing Symmetry of the Integrand
Let $f(x) = \cos^5 x$. We evaluate $f(2\pi - x)$:
$f(2\pi - x) = \cos^5(2\pi - x)$
Since $\cos(2\pi - \theta) = \cos \theta$ [By the Quadrant Rule for Trigonometric Functions], we have:
$f(2\pi - x) = (\cos x)^5 = \cos^5 x = f(x)$
Thus, $I = 2 \int_{0}^{\pi} \cos^5 x \, dx$.
Step 3: Further Symmetry Decomposition
Now consider the integral $\int_{0}^{\pi} \cos^5 x \, dx$. We test the property $\int_{0}^{a} f(x) \, dx = 0$ if $f(a-x) = -f(x)$.
Let $f(x) = \cos^5 x$ and $a = \pi$:
$f(\pi - x) = \cos^5(\pi - x)$
Since $\cos(\pi - x) = -\cos x$ [By the Supplementary Angle Identity], we have:
$f(\pi - x) = (-\cos x)^5 = -\cos^5 x = -f(x)$
Since $f(\pi - x) = -f(x)$, it follows that $\int_{0}^{\pi} \cos^5 x \, dx = 0$.
Step 4: Logical Conclusion
Substituting the result from Step 3 back into the expression derived in Step 2:
$I = 2 \times 0 = 0$
Visual Representation of the Function $\cos^5 x$
Final Solution: $\int_{0}^{2\pi} \cos^5 x \, dx = 0$
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. $\int_{0}^{\frac{\pi}{2}} \frac{\cos^5 x dx}{\sin^5 x + \cos^5 x}$
- Q5: By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19. 5. $\int_{-5}^{5} |x+2| dx$
- 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. $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^7 x dx$
- 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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