1. Evaluate the given using Integration by Partial Fractions (pls follow the format on the reference: A and B as variables) -6x + 10 dx J p²(x² + 2x + 5)
1. Evaluate the given using Integration by Partial Fractions (pls follow the format on the reference: A and B as variables) -6x + 10 dx J p²(x² + 2x + 5)
1. Evaluate the given using Integration by Partial Fractions (pls follow the format on the reference: A and B as variables) -6x + 10 dx J p²(x² + 2x + 5)
Apply integration by partial fractions (where x is related to A, B, C). I provided reference on the 2nd photo. pls skip if unsure & not willing to answer completely. Thanks!
Transcribed Image Text:1. Evaluate the given using Integration by Partial Fractions (pls follow the format on the reference: A
and B as variables)
-6x + 10
dx
x² (x² + 2x + 5)
Transcribed Image Text:Kindly use this reference to provide complete solutions for 1:
Case 1: Denominator has distinct linear factors
If g(x) = (a)x+ bị)(a2x+ b2).…· (a,x+ b,), where all factors are distinct, then
f(x)
g(x) a1x+ bj ' azx+b2
A
A2
A,
anx+ b,'
for some real numbers Aj,...,A,.
Case 2: Denominator has repeated linear factors
If (ax+ b)", n> 1, is a factor of the denominator, then corresponding to this
factor is the sum of n partial fractions
A
A2
ax+ b' (ax+ b)²
A,
(ax+ b)"'
for some real numbers Aj,..,An.
Case 3: Denominator has an irreducible quadratic factor
If ax + bx+ c, a+0, is an irreducible quadratic factor of the denominator
that is not repeated , then the corresponding partial fraction is of the form
Ax+ B
ax² + bx+ c'
for some real numbers Aand B.
If (ax? + bx + c)", n>1 is a factor of the denominator, then corresponding to
this factor is the sum of n partial fractions
A1x+ B1
ax² + bx + c ° (ax² + bx+ c)²
Azx+ B2
Anx+ Bn
(ax² + bx+ c)"
for some real numbers Aj..An, B1..,Bp.
If ax + bx+ c, a#0, is an irreducible quadratic factor of the denominator
that is not repeated, then the corresponding partial fraction is of the form
Ax+ B
ax² + bx+ c'
for some real numbers A and B.
If (ax? + bx + c)", n >1 is a factor of the denominator, then corresponding to
this factor is the sum of n partial fractions
A1x+ BỊ
ax² + bx+ c' (ax + bx+ c)²
for some real numbers Aj,..,A, B1,..,Bp-
A2x + B2
Anx + Bn
+...+
(ax² + bx + c)"
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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