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

### Question 23.

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 4 and f(x) = x + e2x.

=> h = 4/n

=> nh = 4

So, we get,

I =

=

=

=

=

=

Now if h âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’ a

So, we get,

I =

=

=

=

Now if h âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’ x.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

Now if h âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. 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 âˆ’> 0, then n âˆ’> âˆž. So, we have,

=

=

= 8 + 12 + 8

= 28

Therefore, the value ofas limit of sum is 28.

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