Class 12 RD Sharma Solutions - Chapter 19 Indefinite Integrals - Exercise 19.28

Question 1. Find 

Solution:

Let considered x - 1 = t, 

so that dx = dt

Thus, 

Question 2. Evaluate 

Solution:

Let I = 

Question 3. Evaluate 

Solution:

I = 

Hence, 

Question 4. Evaluate 

Solution:

Let I = 

Therefore, I = 

Question 5. 

Solution:

I = 

Let us considered sinx = t

So, on differentiating, we get

cosx dx = dt

I = 

Therefore, I = 

Question 6. Evaluate 

Solution:

I = 

Let us considered e^x = t

So, on differentiating, we get

e^xdx = dt

Therefore, I = 

Hence, I = 

Question 7. Evaluate 

Solution:

I = 

We already have, 

Therefore, I = 

Question 8. Evaluate 

Solution:

Let us assume I = 

Therefore, I = 

Question 9. Evaluate 

Solution:

Let us assume I = 

Therefore, I = 

Question 10. Evaluate 

Solution:

Let us assume I = 

Therefore, I = 

Question 11. Evaluate 

Solution:

Let us assume I = 

Therefore, I = 

Question 12. Evaluate 

Solution:

Let us assume x² = t

On differentiating we get

2x dx = dt

Therefore, I = 

Hence, I = 

Question 13. Evaluate 

Solution:

I = 

Let us considered x³ = t

So, on differentiating, we get

3x²dx = dt

Therefore, I = 

Hence, I = 

Question 14. Evaluate 

Solution:

I = 

Let us considered logx = t

So, on differentiating, we get

1/x dx = dt 

Therefore, I = 

Hence, I = 

Question 15. Evaluate 

Solution:

I = 

Therefore, I = 

Question 16. Evaluate 

Solution:

Let I = 

I = 

Summary

Exercise 19.28 in RD Sharma's Class 12 Chapter 19 on Indefinite Integrals focuses on integrating rational functions where the denominator is a fourth-degree polynomial (quartic). The key techniques used in solving these integrals include:

• Partial fraction decomposition
• Substitution method
• Recognizing standard integral forms
• Using trigonometric substitutions
• Long division of polynomials (when degree of numerator ≥ degree of denominator)

These problems require a strong understanding of algebraic manipulation, factorization of polynomials, and various integration techniques. Students should be comfortable with complex fractions and identifying the most efficient method for each problem.