Home
Class 12
MATHS
For every function f (x) which is twice ...

For every function f (x) which is twice differentiable , these will be good approximation of
`int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)}`, for more acutare results for ` cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)]`
When ` c= (a+b)/(2)`
`int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx`
If `lim_(t toa) (int_(a)^(t)f(x)dx-((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0`, then degree of polynomial function f (x) atmost is

A

0

B

1

C

3

D

2

Text Solution

Verified by Experts

The correct Answer is:
B
`int_(0)^(pi//2)sin x dx =((pi)/(2)-0)/(4)[sin +sin ((pi)/(2))+2 sin((0+(pi)/(2))/(2))]`
`=(pi)/(8)(1+sqrt(2))`
Promotional Banner

Similar Questions

Explore conceptually related problems

int_(a)^(b) f(x) dx =

Prove that int_(a)^(b)f(x)dx=(b-a)int_(0)^(1)f((b-a)x+a)dx

Given a real-valued function f which is monotonic and differentiable. Then int_(f(a))^(f(b))2x(b-f^(-1)(x))dx=

Evaluate int_(0)^(a)(f(x))/(f(x)+f(a-x)) dx.

Let the definite integral be defined by the formula int_(a)^(b)f(x)dx=(b-a)/2(f(a)+f(b)) . For more accurate result, for c epsilon (a,b), we can use int_(a)^(b)f(x)dx=int_(a)^(c)f(x)dx+int_(c)^(b)f(x)dx=F(c) so that for c=(a+b)/2 we get int_(a)^(b)f(x)dx=(b-a)/4(f(a)+f(b)+2f(c)) . If f''(x)lt0 AA x epsilon (a,b) and c is a point such that altcltb , and (c,f(c)) is the point lying on the curve for which F(c) is maximum then f'(c) is equal to

If f(a+b-x)=f(x) , then int_(a)^(b)xf(x)dx is equal to

If f(x) is monotonic differentiable function on [a , b] , then int_a^bf(x)dx+int_(f(a))^(f(b))f^(-1)(x)dx= (a) bf(a)-af(b) (b) bf(b)-af(a) (c) f(a)+f(b) (d) cannot be found

If int_(0)^(x) f(t)dt=x^2+int_(x)^(1) t^2f(t)dt , then f'(1/2) is

If f(x) is differentiate in [a,b], then prove that there exists at least one c in (a,b)"such that"(a^(2)-b^(2))f'(c)=2c(f(a)-f(b)).

If f(x)=f(a+x) then show that int_(0)^(2a)f(x)dx=2int_(0)^(a)f(x)dx .