Question

Integration by Parts and Partial Fraction Decomposition

Answer 1

you have to combine those ugly fractions and compare the tops on both sides of the equations to determine A, B, and C

Answer 2

\[38-3t^2-25t=A \left( t-1 \right)\left( t-2 \right)+B \left( t+4 \right)\left( t-2 \right)+C \left( t+4 \right)\left( t-1 \right)\]
comparing co-efficients of like powers
-3=A+B+C
-25=-3A+2B+3C
38=2A-8B-4C
solve for A,B,C

Answer 3

A=3

Answer 4

B=-2, C= -4

Answer 5

or \[\frac{ 38-3t^2-25t }{ (t+4)(t-1)(t-2) }=\frac{ 38-3(-4)^2-25(-4) }{ (t+4)(-4-1)(-4-2) }+?\]

Answer 6

have you calculated A,B&C

Answer 7

yes

Answer 8

as stated above

Answer 9

\[\int\limits \frac{ 3 }{ t+4 }dt=3\ln \left( t+4 \right)=\ln \left( t+4 \right)^3\]
similarly solve other two.

Answer 10

\[\int\limits \frac{ -2 }{ t-1 }dt=-2\ln \left| t-1 \right|=-\ln \left| t-1 \right|^2=-\ln \left( t-1 \right)^2\]

Answer 11

similarly third