# L-Hospitalâ€™s rule

If at *x* = *a*, a function *f*(*x*) has an indeterminate value such as etc. then L-hospitalâ€™s rule can be applied.

consider two functions *f*(*x*) and Ï•(*x*) having value 0 at *x* = 0, then

If still *f *â€² (*x*) and Ï•â€² (*x*) has value zero at *x* = *a*, then and goes on.

# Mean Value Theorems

**Rolleâ€™s theorem**

If a function*f*(*x*) is continuous in a closed interval [*a*,*b*] i.e.*a*â‰¤*x*â‰¤*b*, and derivable in open interval (*a*,*b*) i.e.*a*<*x*<*b*, and if*f*(*a*) =*f*(*b*), then there exist atleast one real*c*in (*a*,*b*) such that*f*â€²(*c*) = 0**Lagrangeâ€™s first mean value theorem**

If a function*f*(*x*) is continuous in closed interval [*a*,*b*] and*fâ€™(x)*exist in open interval (a, b) then there exist atleast one value of c such that,

*f*â€²*c*)

# Standard Integration

- ; (
*a*= constant) - when n = 1, i.e.,

*u* is first function, *v* is second function. To find first function out of two we follow â€˜**ILATE**â€™ rule.

I-Inverse, L-Log function, A-Algebraic funtion,

T- Trignomatric function, E-Exponential function.