Click to Chat

1800-2000-838

+91-120-4616500

CART 0

• 0

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping
• Complete JEE Main/Advanced Course and Test Series
• OFFERED PRICE: Rs. 15,900
• View Details
Get extra Rs. 3,180 off
USE CODE: CHEM20

```Solved Examples on Definite Integral

1. 5/4                                                                                            2. 7
3. 4                                                                                                4. 2

Solution: Given that F(x) = ∫ f(t) dt, where integral runs from 0 to x.

By Leibnitz rule, we have

F’(x) = f(x)

But F(x2) = x2 (1+x) = x2 + x3

Hence, this gives F(x) = x + x3/2

So, F’(x) = 1+ 3/2 x1/2

Hence, f(x) = F’(x) = 1+ 3/2 x1/2

Hence, f(4) = 1+ 3/2 . (4)1/2

This gives f(4) = 1+ 3/2 . 2 = 4

Hence, option (3) is correct.

1. ±1                                                             2. ±1/ √2

3. ±1/ 2                                                         4. 0 and 1

Solution: F(x) = ∫√2-t2 dt, where the integral runs from 1 to x

Hence, f’(x) = √2-x2

Another condition that is given in the question is that x2 – f’(x) = 0

Therefore, x2 = √2-x2

Hence, x4 = 2-x2

So, x4 + x2 – 2 = 0

This gives the value of x as ±1.

1. f(1/2) < 1/2 and f(1/3) > 1/3

2. f(1/2) > 1/2 and f(1/3) > 1/3

3. f(1/2) < 1/2 and f(1/3) < 1/3

4. f(1/2) > 1/2 and f(1/3) < 1/3

Solution: The given condition is that ∫ √1- (f’(t))2 dt = ∫ f(t) dt , where integral runs from 0 to x and 0 ≤ x ≤ 1.

Differentiating both sides with respect to x by using the Leibnitz rule, we get

√1- (f’(x))2 = f(x)

This gives f’(x) = ± √1- (f (x))2

∫ [f’(x) / √1- (f (x))2] dx = ± ∫ dx

This yields, sin-1 (f(x)) = ± x + c

Now, putting x = 0 we get,

sin-1 (f(0)) = c

Hence, since f(0) = 0, so we have c = sin-1 (0) = 0

Therefore, f(x) = ± sin x

But f(x) ≥ 0, ∀ x ∈ [0, 1]

Hence, f(x) = sin x

As we know that, sin x < x ∀ x > 0

Hence, sin (1/2) < 1/2 and sin (1/3) < 1/3

This gives, f (1/2) < 1/2 and f (1/3) < 1/3.

Hence, the correct option is (3).