
Integral
calculus 

The
indefinite integral 
Integration by parts rule 
The indefinite integrals
containing quadratic polynomial (trinomial)

The indefinite integrals
containing quadratic polynomial, examples







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.



Use similar methods to solve this
integral as in the preceding example. Hence,
the solutions depend on the
sign of the leading
coefficient
a
and the sign or the value of the vertical translation y_{0}.


Let solve given integral by separating
the derivative of the quadratic polynomial in the numerator.




Use the substitution mx + n = 1/ t .






The integrand function to be real, the
quadratic polynomial must be positive, therefore

a
) if a
> 0 and
y_{0}
is not
0 then


see the solutions of the integrals,
example 31 and 36. By using above substitutions


Note that the sign of the vertical
translation y_{0}
affects the solution, i.e.,
changes the sign of its second term.

b
) if a
< 0 and
y_{0}
>
0 then


see the solution of the integral
example 8 above. After applying the substitutions











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