
Integral
calculus 


Substitution
and definite integration

Substitution
and definite integration examples

Integrations
by parts and the definite integral






Substitution
and definite integration

When
we evaluate an indefinite integral of the form


we
use the substitutions g(x)
= u and g'(x) dx = du
to obtain


Therefore,
to evaluate the definite integral using substitution


or


Note,
by making substitution the definite integral changes the limits of
integration. That is, the new limits of integrations
must correspond to the range of values for u
as we now integrate the simpler integrand f
(u)
from u_{a}
= g (a) to
u_{b} = g
(b).


Substitution
and definite integration, examples

Example:
Evaluate 







Integrations
by parts and the definite integral

If
u(x)
and v(x)
are continuous functions with continuous derivatives on [a,
b], then














Calculus contents
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