(with the degree of the
numerator smaller than the degree of the denominator) has the partial
fraction expansion
= 
+ 
.
The goal of this problem is to determine A and B using
two different methods, as described below.
- Multiply this expression through by the denominator on the left
and simplify (cancel) so that there are no fractions left, BUT
DO NOT MULTIPLY OUT. Call your resulting equation (1a).
- By making clever choices for x (two different
choices), determine formulas for A and B in
terms of a and b. (This is the first method for
determining A and B.)
- Return to Equation (1a), multiply out the right hand side, and
collect like terms (in powers of x).
This gives you a new equation, ax + b = … . Write out this equation and call it equation (1c).
- By comparing coefficients in Equation (1c), derive two equations relating a and b to A and B, and solve these equations for A and B. (This is the second method for determining A and B.)
Assume that
x
eax dx = Ax
eax + B
eax, for some unknown A and
B.The goal of this problem is to determine A and B using two different methods, as described below.
- i. Find the derivative of the right hand side.
ii. Use the definition of antiderivative to obtain an equation relating the derivative which you just obtained to the integrand on the left. Call your resulting equation (2a).
- Make a simple, clever choice for x which allows you
to determine a formula for A + Ba .
Now choose another value for x (e.g., x = 1), to obtain another equation. Solve the two equations for A and B in terms of a. (This is the first method for determining A and B.)
- Return to Equation (2a), cancel the common exponential factor
(if you have not done so already), and collect like terms (in
powers of x). This gives you a (possibly) new equation.
Write out this equation and call it equation (2c).
- By comparing coefficients in Equation (2c), obtain two equations relating a, A, and B, and solve these equations for A and B. (This is the second method for determining A and B.)