Let f ( x ) be a function defined on the interval [ a , b ] and F ( x ) be its antiderivative . Then ,
The above is called the second fundamental theorem of calculus .
is defined as the definite integral of f ( x ) from x = a to x = b . The numbers and b are called limits of integration . We write
Evaluation of Definite Integrals by Substitution Consider a definite integral of the following form
Step 1 Substitute g ( x ) = t
⇒ g ‘( x ) dx = dt
Step 2 Find the limits of integration in new system of variable i . e .. the lower limit is g ( a ) and the upper limit is g ( b ) and the g ( b ) integral is now
Step 3 Evaluate the integral , so obtained by usual method .