# 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) = x

^{2}+ 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) = x

^{2}+ 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) = 2x

^{2}+ 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) = x

^{2}.=> 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) = x

^{2}+ 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) = x

^{2}− 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) = 2x

^{2}+ 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) = 3x

^{2}+ 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.