6.6 Techniques of Integration
(Integration by Substitution)
It is often possible to reduce the
complexity of an integration problem by
making an appropriate change of
variable.
Let f(x) and g(x) be two given functions
and let F(x) be an antiderivative for
f(x).The chain rule asserts
Turning this formula into an integration
formula, we have
where C is any constant.
If we set f(x) = x3 and g(x) = x2 + 1, then
f(g(x)) = (x2 + 1)3 and g’(x) = 2x.
By applying aforementioned formula, an
antiderivative F(x) of f(x) is given by
Now, we have
Let us now consider the problem again by
using a substitution.
Replace g(x) by a new variable u, and
replace g’(x) dx with du.
We reduce the complexity of f(g(x)) to
f(u).
Now,
Once the antiderivative F(u) is computed,
then we make the substitution of g(x) for
u and we have
Then,
The complex integral becomes a simpler one
We replace u with g(x) and we are done
Integration by Substitution
1.Define a new variable u = g(x), where g(x) is
chosen in such a way that, when written in
terms of u, the integrand is simpler than
when written in terms of x.
2.Transform the integral with respect to x into
an integral with respect to u by replacing g(x)
everywhere by u and g’(x) dx by du.
3.Integrate the resulting function of u.
4.Rewrite the answer in terms of x by replacing
u by g(x).