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| Integral
calculus |
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The
indefinite integral |
Integration by parts rule |
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The indefinite integrals
containing quadratic polynomial (trinomial)
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The indefinite integrals
containing quadratic polynomial, examples
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Integration
by parts rule
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The rule for differentiating the
product of two differentiable functions leads to the integration
by parts formula.
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Let f
(x)
and g
(x)
are differentiable functions, then the product rule gives
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[ f
(x) g (x)]'
= f (x)
g (x)' + g (x)
f ' (x),
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integrating both sides |
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Since the integral of the derivative
of a function is the function itself, then
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and by rearranging obtained is
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the integration by parts
formula.
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By substituting
u = f
(x) and
v
= g (x)
then, du =
f ' (x)
dx and dv
= g' (x)
dx, so that
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To apply the above formula, the
integrand of a given integral should represent the product of one
function and the differential of the other.
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The selection of the function
u and the differential dv
should simplify the
evaluation of the remaining integral.
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In some cases it will be necessary to
apply the integration by parts repeatedly to obtain a simpler
integral.
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The indefinite integrals
containing quadratic polynomial (trinomial)
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that the quadratic polynomial y
= ax2 + bx + c
represents the expansion of the translatable form of
its source function y
= ax2 in the direction of the coordinate axes, thus
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We solve given integral by separating
the derivative of the quadratic polynomial in the numerator, thus
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therefore, the solution of I2
depend on the leading coefficient a
and the vertical translation y0,
i.e.,
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)
if |
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then, |
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) if |
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then, |
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c
) if
y0 = 0
then,
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The integrand function to be real, the
quadratic polynomial in the denominator must be positive,
therefore
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a
) if a
> 0 and
y0
is not
0 then
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b
) if a
< 0 and
y0
>
0 then
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c
) if a
> 0 and
y0 =
0 then
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| Functions
contents G |
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