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7. Integration by Parts
by M. Bourne
If u and v are functions of x, the product rule for differentiation that we met earlier
gives us:
Rearranging, we have:
Integrating throughout, with respect to x, we obtain the formula for integration by
parts:
This formula allows us to turn a complicated integral into more simple ones. We must
make sure we choose u and dv carefully.
Function u is chosen so that
is simpler than u.
Priorities for Choosing u
1. Let u = ln x
2. Let u = xn
3. Let u = enx
Example 1:
Solution:
We could let u = x or u = sin 2x. In general, we choose the one that allows
of a simpler form than u.
So for this example, we choose u = x and dv = sin 2x dx.
u=x
dv = sin 2x dx
du = dx
Substituting into the integration by parts formula, we get:
to be
Example 2:
We could let u = x or
.
Once again, we choose the one that allows
choose u = x.
to be of a simpler form than u, so we
Therefore du = dx. It follows that dv must be given by:
u=x
du = dx
Substituting into the integration by parts formula, we get:
.
Example 3:
We could let u = x2 or
.
Considering the priorities given above, we choose u = ln 4x and so we must let dv =
x2 dx.
u = ln 4x
Substituting, we get:
dv = x2 dx
Example 4:
u=x
du = dx
v = tan x
Example 5:
u = x2
dv = e-x dx
du = 2x dx
v = -e-x
Now, the integral we are left with cannot be found immediately. We need to perform
integration by parts again, for this integral.
u=x
dv = e-x dx
du = dx
v = -e-x
So putting this answer together with the answer for the first part, we have the final
solution:
Example 6:
u = ln x
dv = dx
v=x
Integration by Parts
From the product rule, we can obtain the following formula, which is very useful in
integration:
It is used when integrating the product of two expressions (a and b in the bottom
formula). When using this formula to integrate, we say we are "integrating by parts".
Sometimes you will have to integrate by parts twice (or possibly even more times)
before you get an answer.
Example
Find ∫xe-x dx
Integrating by parts (with v = x and du/dx = e-x), we get:
-xe-x - ∫-e-x dx
(since ∫e-x dx = -e-x)
= -xe-x - e-x + constant
We can also sometimes use integration by parts when we want to integrate a function
that cannot be split into the product of two things. The trick we use in such
circumstances is to multiply by 1 and take du/dx = 1.
Example
Find ∫ ln x dx
To integrate this, we use a trick, rewrite the integrand (the expression we are
integrating) as 1.lnx . We then let v = ln x and du/dx = 1 .
Hence ∫ ln x dx = x ln x - ∫ x (1/x) dx
= x lnx - ∫ dx
= x lnx - x + constant
Integration by Parts
Integration by parts is an integration technique which is based on the Product Rule for
derivatives. The Product Rule says:
Integrate both sides, then rearrange the equation:
The last line gives the Integration by Parts formula: