
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.


The indefinite integrals
containing quadratic polynomial (trinomial)

Recall
that the quadratic polynomial y
= ax^{2} + bx + c
represents the expansion of the translatable form of
its source function y
= ax^{2} in the direction of the coordinate axes, thus




We solve given integral by separating
the derivative of the quadratic polynomial in the numerator, thus


therefore, the solution of I_{2}
depend on the leading coefficient a
and the vertical translation y_{0},
i.e.,

a
)
if 

then, 



b
) if 

then, 



c
) if
y_{0 }= 0
then,





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

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


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


c
) if a
> 0 and
y_{0} =
0 then











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