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Q5:
Find the integrals of the functions in Exercises 1 to 22:
5. $\sin^3 x \cos^3 x$
Solution :
Step 1: Establishing the Integral Expression
The given integral is: $I = \int \sin^3 x \cos^3 x \, dx$.
Step 2: Decomposition of the Integrand
To facilitate the method of substitution, we decompose the power functions while keeping one cosine term for the differential $du$.
$I = \int \sin^3 x \cdot \cos^2 x \cdot \cos x \, dx$ [Using index laws $a^{m+n} = a^m \cdot a^n$]
Utilize the trigonometric identity $\cos^2 x = 1 - \sin^2 x$ [Per the Pythagorean Identity $\sin^2 x + \cos^2 x = 1$]:
$I = \int \sin^3 x (1 - \sin^2 x) \cos x \, dx$
Step 3: Variable Substitution
Let $u = \sin x$.
Differentiating both sides with respect to $x$ gives: $\frac{du}{dx} = \cos x$, which implies $du = \cos x \, dx$ [By the Chain Rule of differentiation].
Step 4: Algebraic Transformation of the Integral
Substituting $u$ and $du$ into the integral expression:
$I = \int u^3 (1 - u^2) \, du$
$I = \int (u^3 - u^5) \, du$ [By the distributive property of multiplication over addition/subtraction]
Step 5: Applying the Power Rule for Integration
Apply the Power Rule $\int u^n \, du = \frac{u^{n+1}}{n+1} + C$:
$I = \int u^3 \, du - \int u^5 \, du$
$I = \frac{u^4}{4} - \frac{u^6}{6} + C$ [Where $C$ is the constant of integration]
Step 6: Back-Substitution
Replace $u$ with $\sin x$ to return to the original variable:
$I = \frac{\sin^4 x}{4} - \frac{\sin^6 x}{6} + C$
Analytical Visual Representation of the Substitution Strategy
Final Solution: $\frac{\sin^4 x}{4} - \frac{\sin^6 x}{6} + C$
More Questions from Class 12 Mathematics Integrals EXERCISE 7.3
- Q1: Find the integrals of the functions in Exercises 1 to 22: 1. $\sin^2(2x+5)$
- Q2: Find the integrals of the functions in Exercises 1 to 22: 2. $\sin 3x \cos 4x$
- Q3: Find the integrals of the functions in Exercises 1 to 22: 3. $\cos 2x \cos 4x \cos 6x$
- Q4: Find the integrals of the functions in Exercises 1 to 22: 4. $\sin^3(2x+1)$
- Q6: Find the integrals of the functions in Exercises 1 to 22: 6. $\sin x \sin 2x \sin 3x$
- Q7: Find the integrals of the functions in Exercises 1 to 22: 7. $\sin 4x \sin 8x$
- Q8: Find the integrals of the functions in Exercises 1 to 22: 8. $\frac{1-\cos x}{1+\cos x}$
- Q9: Find the integrals of the functions in Exercises 1 to 22: 9. $\frac{\cos x}{1+\cos x}$
- Q10: Find the integrals of the functions in Exercises 1 to 22: 10. $\sin^4 x$
- Q11: Find the integrals of the functions in Exercises 1 to 22: 11. $\cos^4 2x$
- Q12: Find the integrals of the functions in Exercises 1 to 22: 12. $\frac{\sin^2 x}{1+\cos x}$
- Q13: Find the integrals of the functions in Exercises 1 to 22: 13. $\frac{\cos 2x - \cos 2\alpha}{\cos x - \cos \alpha}$
- Q14: Find the integrals of the functions in Exercises 1 to 22: 14. $\frac{\cos x - \sin x}{1+\sin 2x}$
- Q15: Find the integrals of the functions in Exercises 1 to 22: 15. $\tan^3 2x \sec 2x$
- Q16: Find the integrals of the functions in Exercises 1 to 22: 16. $\tan^4 x$
- Q17: Find the integrals of the functions in Exercises 1 to 22: 17. $\frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$
- Q18: Find the integrals of the functions in Exercises 1 to 22: 18. $\frac{\cos 2x + 2\sin^2 x}{\cos^2 x}$
- Q19: Find the integrals of the functions in Exercises 1 to 22: 19. $\frac{1}{\sin x \cos^3 x}$
- Q20: Find the integrals of the functions in Exercises 1 to 22: 20. $\frac{\cos 2x}{(\cos x + \sin x)^2}$
- Q21: Find the integrals of the functions in Exercises 1 to 22: 21. $\sin^{-1}(\cos x)$
- Q22: Find the integrals of the functions in Exercises 1 to 22: 22. $\frac{1}{\cos(x-a)\cos(x-b)}$
- Q23: Choose the correct answer in Exercises 23 and 24. 23. $\int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx$ is equal to
- Q24: Choose the correct answer in Exercises 23 and 24. 24. $\int \frac{e^x(1+x)}{\cos^2(e^x x)} dx$ equals
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 7.4
- 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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