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Integral
calculus |
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The
indefinite integral |
The indefinite integrals
containing quadratic polynomial (trinomial)
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The indefinite integrals
containing quadratic polynomial, examples
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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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a
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if |
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then, |
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b
) 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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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 y0.
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Let solve given integral by separating
the derivative of the quadratic polynomial in the numerator.
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Use the substitution
mx + n = 1/ t .
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The integrand function to be real, the
quadratic polynomial must be positive, therefore
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a
) if a
> 0 and
y0
is not
0 then
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see the solutions of the integrals,
example 31 and 36. By using above substitutions
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Note that the sign of the vertical
translation y0
affects the solution, i.e.,
changes the sign of its second term.
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b
) if a
< 0 and
y0
>
0 then
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see the solution of the integral
example 8 above. After applying the substitutions
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Calculus contents
F |
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