
Integral
calculus 


The
indefinite integral 
Integration by parts rule 
Evaluating the indefinite integrals using the integration by parts
formula, examples

Evaluating the indefinite integrals using the integration by parts
formula, solutions






Integration
by parts rule

The rule for differentiating the
product of two differentiable functions leads to the integration
by parts formula.

Let f
(x)
and g
(x)
are differentiable functions, then the product rule gives

[ f
(x) g (x)]'
= f (x)
g (x)' + g (x)
f ' (x),

by
integrating both sides 

Since the integral of the derivative
of a function is the function itself, then


and by rearranging obtained is


the integration by parts
formula.

By substituting
u = f
(x) and
v
= g (x)
then, du =
f ' (x)
dx and dv
= g' (x)
dx, so that


To apply the above formula, the
integrand of a given integral should represent the product of one
function and

the differential of the other.

The selection of the function
u and the differential dv
should simplify the
evaluation of the remaining integral.

In some cases it will be necessary to
apply the integration by parts repeatedly to obtain a simpler
integral.


Evaluating the indefinite integrals using the integration by parts
formula, examples

Evaluate
the following indefinite integrals using the integration by parts
formula;

41. 

42. 


43. 

44. 


45. 

46. 



Evaluating the indefinite integrals using the integration by parts
formula, solutions

Example:
41.
Evaluate 




Example:
42.
Evaluate 




Example:
43. Evaluate 




Example:
44.
Evaluate 




Example:
45. Evaluate 




Example:
46. Evaluate 












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