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
ā gā( 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.