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Chapter 28: Methods of Integration
Sec 7: Integration by Parts
There are many methods of transforming integrals into forms that can be integrated by one of th
basic formulas.
Since the derivative of a product of functions is found by use of the formula
The differential of a product of functions is given by
. Integrating both sides of this equation, we
have. Solving for
, we obtain
Example 1
Integrate:
Integral doesn’t fit any of previous forms we discussed. Neither x nor sin x can be made a factor
of a proper du. However, by choosing
u = x
and dv = sin x dx, integration by parts can be used.
Finding the differential of u and integrating dv, we find du and v. this gives us
Now we substitute to Eq. 1
Example 2:
Integrate
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View Full DocumentForm doesn’t fit general power rule, for x dx is not factor of differential of 1 – x. Choosing
u = x
and , we have
and v can readily be determined.
Substitute to Eq. 1
This point, see we can complete integration.
1.
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