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Class 12 RD Sharma Solutions – Chapter 20 Definite Integrals – Exercise 20.5 | Set 1

• Last Updated : 20 May, 2021

Question 1.

Solution:

We have,

I =

We know,, where h =

Here a = 0, b = 3 and f(x) = x + 4.

=> h = 3/n

=> nh = 3

So, we get,

I =

=

=

=

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

=

=

= 12 +

=

Therefore, the value ofas limit of sum is.

Question 2.

Solution:

We have,

I =

We know,

, where h =

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

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

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

=

=

= 6 + 2

= 8

Therefore, the value ofas limit of sum is 8.

Question 3.

Solution:

We have,

I =

We know,

, where h =

Here a = 1, b = 3 and f(x) = 3x âˆ’ 2.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

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

=

=

= 2 + 6

= 8

Therefore, the value ofas limit of sum is 8.

Question 4.

Solution:

We have,

I =

We know,

, where h =

Here a = âˆ’1, b = 1 and f(x) = x + 3.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

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

=

=

= 4 + 2

= 6

Therefore, the value ofas limit of sum is 6.

Question 5.

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 6.

Solution:

We have,

I =

We know,

, where

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

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

=

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

=

=

= 10 + 4

= 14

Therefore, the value ofas limit of sum is 14.

Question 7.

Solution:

We have,

I =

We know,

, where h =

Here a = 3, b = 5 and f(x) = 2 âˆ’ x.

=> h = 2/n

=> nh = 2

So, we get,

I =

=

=

=

=

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

=

=

= â€“2 â€“ 2

= â€“4

Therefore, the value ofas limit of sum is â€“4.

Question 8.

Solution:

We have,

I =

We know,

, where h =

Here a = 0, b = 2 and f(x) = x2 + 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 9.

Solution:

We have,

I =

We know,

, where h =

Here a = 1, b = 2 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,

=

=

= 1 + 1 +

= 1 + 1 +

=

Therefore, the value ofas limit of sum is.

Question 10.

Solution:

We have,

I =

We know,

, where h =

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

=> h = 1/n

=> nh = 1

So, we get,

I =

=

=

=

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

=

=

= 9 + 4 +

=

Therefore, the value ofas limit of sum is.

Question 11.

Solution:

We have,

I =

We know,

, where h =

Here a = 1, b = 2 and f(x) = x2 âˆ’ 1.

=> h = 1/n

=> nh = 1

So, we get,

I =

=

=

=

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

=

=

= 1 +

= 1 +

=

Therefore, the value of as limit of sum is .

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