Integration of product of two functions
Answers
Refer to the image attached, You will understand it.
Answer:
Step-by-step explanation:
We already know how to differentiate a product: if
y = u v
then
dy
dx
=
d(uv)
dx
= u
dv
dx
+ v
du
dx
.
Rearranging this rule:
u
dv
dx
=
d(uv)
dx
− v
du
dx
.
Now integrate both sides:
Z
u
dv
dx
dx =
Z
d(uv)
dx
dx −
Z
v
du
dx
dx .
The first term on the right simplifies since we are simply integrating what has been differentiated.
Z
u
dv
dx
dx = u v −
Z
v
du
dx
dx .
This is the formula known as integration by parts.
Key Point
Integration by parts
Z
u
dv
dx
dx = u v −
Z
v
du
dx
dx
The formula replaces one integral (that on the left) with another (that on the right); the intention
is that the one on the right is a simpler integral to evaluate, as we shall see in the following
examples.
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3. Using the formula for integration by parts
Example
Find Z
x cos x dx.
Solution
Here, we are trying to integrate the product of the functions x and cos x. To use the integration
by parts formula we let one of the terms be dv
dx
and the other be u. Notice from the formula that
whichever term we let equal u we need to differentiate it in order to find du
dx
. So in this case, if
we let u equal x, when we differentiate it we will find du
dx
= 1, simply a constant. Notice that
the formula replaces one integral, the one on the left, by another, the one on the right. Careful
choice of u will produce an integral which is less complicated than the original.
Choose
u = x and dv
dx
= cos x .
With this choice, by differentiating we obtain
du
dx
= 1 .
Also from dv
dx
= cos x, by integrating we find
v =
Z
cos x dx = sin x .
(At this stage do not concern yourself with the constant of integration). Then use the formula
Z
u
dv
dx
dx = u v −
Z
v
du
dx
dx :
Z
x cos x dx = x sin x −
Z
(sin x) · 1 dx
= x sin x + cos x + c
where c is the constant of integration.
In the next Example we will see that it is sometimes necessary to apply the formula for integration
by parts more than once.
Example
Find Z
x
2
e
3x
dx.
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