Class 12 RD Sharma Solutions - Chapter 20 Definite Integrals - Exercise 20.5 | Set 3

Evaluate the following definite integrals as limits of sums:

Question 23. 

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 4 and f(x) = x + e^2x.

=> h = 4/n

=> nh = 4

So, we get,

I =

=

=

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

=

=

Therefore, the value ofas limit of sum is.

Question 24. 

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 2 and f(x) = x² + x.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

=

=

Therefore, the value ofas limit of sum is.

Question 25. 

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 2 and f(x) = x² + 2x + 1.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

=

=

Therefore, the value ofas limit of sum is.

Question 26. 

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 3 and f(x) = 2x² + 3x + 5.

=> h = 3/n

=> nh = 3

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

= 15 + 18 +

=

Therefore, the value ofas limit of sum is.

Question 27. 

Solution:

We have,

I =

We know,

, where h =

Here a = a, b = b and f(x) = x.

=> h =

=> nh = b − a

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

=

=

=

=

Therefore, the value ofas limit of sum is.

Question 28. 

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 5 and f(x) = x + 1.

=> h =5/n

=> nh = 5

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

= 5 +

=

Therefore, the value ofas limit of sum is.

Question 29. 

Solution:

We have,

I =

We know,

, where h =

Here a = 2, b = 3 and f(x) = x².

=> h = 1/n

=> nh = 1

So, we get,

I =

=

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

=

=

=

Therefore, the value ofas limit of sum is.

Question 30. 

Solution:

We have,

I =

We know,

, where h =

Here a = 1, b = 3 and f(x) = x² + x.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

=

=

Therefore, the value ofas limit of sum is.

Question 31. 

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 2 and f(x) = x² − x.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

=

=

Therefore, the value ofas limit of sum is.

Question 32. 

Solution:

We have,

I =

We know,

, where h =

Here a = 1, b = 3 and f(x) = 2x² + 5x.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

= 14 + 18 +

=

Therefore, the value ofas limit of sum is.

Question 33. 

Solution:

We have,

I =

We know,

, where h =

Here a = 1, b = 3 and f(x) = 3x² + 1.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

Now if h −> 0, then n −> ∞. So, we have,

=

=

= 8 + 12 + 8

= 28

Therefore, the value ofas limit of sum is 28.

Practice Questions on Definite Integrals

1. Evaluate: ∫[0 to π/2] (sin x)/(1 + cos x) dx

2. Calculate: ∫[0 to 1] x^2 / (1 + x^3) dx

3. Find the value of: ∫[0 to π/4] tan x dx

4. Evaluate: ∫[0 to 1] dx / (1 + x^2)^2

5. Compute: ∫[0 to π/2] cos^3 x dx

6. Calculate: ∫[0 to 1] x ln(1 + x) dx

7. Evaluate: ∫[0 to π/2] sin^2 x cos^2 x dx

8. Find the value of: ∫[1 to e] (ln x)^2 dx

9. Compute: ∫[0 to 1] x^2 e^x dx

10. Evaluate: ∫[0 to π/4] sec^3 x dx

Summary

Chapter 20 of RD Sharma's Class 12 mathematics textbook focuses on Definite Integrals. Exercise 20.5 | Set 3 deals with more advanced techniques for evaluating definite integrals. Key points covered in this exercise set include:

• Integration of trigonometric functions and their combinations
• Definite integrals involving algebraic and transcendental functions
• Application of substitution method in definite integrals
• Integration by parts for definite integrals
• Handling integrals with logarithmic and exponential functions
• Techniques for evaluating integrals with rational functions
• Use of trigonometric identities to simplify integrands
• Recognizing and applying standard integral formulas