Evaluate using tabular method:
integral e^(2x)cos3xdx
Let M = ∫ e^(2x) cos3x dx
u = cos3x
du = -3sin3x dx
dv = e^(2x) dx
v = 1/2 e^(2x)
M = uv - ∫v du
M = 1/2 e^(2x) cos3x + (3/2)∫e^(2x) sin3x dx
Now consider ∫e^(2x) sin3x dx
u = sin3x
du = 3cos3x dx
dv = e^(2x) dx
v = 1/2 e^(2x)
∫e^(2x) sin3x dx = 1/2 e^(2x) sin3x - 3/2 ∫e^(2x) cos3x dx
= 1/2 e^(2x) sin3x - 3/2 M
So, now we have
M = 1/2 e^(2x) cos3x + (3/2)[1/2 e^(2x) sin3x - 3/2 M]
M = 1/2 e^(2x) cos3x + 3/4 e^(2x) sin3x - 9/4 M
13/4 M = 1/4 e^(2x) (2cos3x+3sin3x)
M = 1/13 e^(2x) (2cos3x+3sin3x) + C
Note that you could just as well have used
u = e^(2x)
du = 2e^(2x) dx
dv = cos3x dx
v = 1/3 sin3x
and again two integrations by parts would have worked.
Not sure how this would have worked using the tabular method, since there is no nth derivative which is zero.
To evaluate the integral ∫e^(2x)cos(3x)dx using the tabular method (also known as integration by parts or the method of successive differentiation), follow these steps:
Step 1: Choose the function to differentiate and integrate.
Let's label the two functions as "u" and "dv" as follows:
u = e^(2x) (function to differentiate)
dv = cos(3x)dx (function to integrate)
Step 2: Determine the derivatives and antiderivatives of the functions.
Differentiating u once gives:
du/dx = 2e^(2x)
Integrating dv gives:
∫dv = ∫cos(3x)dx = (1/3)sin(3x)
Step 3: Set up the table and fill in the columns.
We create a table with two columns: "u" and "v". Add two additional columns for the derivatives of "u" and the antiderivative of "v".
_____________________________________
| u | v | du/dx | ∫v dx |
-------------------------------------
| | | | |
In the first row, fill in the values for "u" and ∫v dx:
_____________________________________
| e^(2x) | ? | ? | (1/3)sin(3x) |
-------------------------------------
| | | | |
Step 4: Fill in the columns.
In the second row, calculate the derivatives (du/dx) of "u" and the antiderivative of "v" (∫v dx):
_____________________________________
| e^(2x) | ? | 2e^(2x) | (1/3)sin(3x) |
-------------------------------------
| | | | |
Step 5: Determine the patterns in the columns.
By observing the table, we notice a pattern. The derivative of "u" is a multiple of the original "u" (2e^(2x)). The antiderivative of "v" alternates between itself and its negative [(1/3)sin(3x)].
Step 6: Multiply and adjust signs.
Multiply the values in the "u" and "∫v dx" columns, and adjust the signs in alternating rows:
_____________________________________
| e^(2x) | ? | 2e^(2x) | (1/3)sin(3x) |
--------------------------------------
| ? | -1/3 | -2/9 | -(1/9)cos(3x) |
Step 7: Integrate and simplify.
Integrate the multiplied values in the second row:
____________________________________________________
| e^(2x) | -1/3 | -2/9 | -(1/9)cos(3x) |
------------------------------------------------------
| (-1/6)e^(2x) | -1/3x + (2/27)e^(2x) | (1/27)sin(3x) - (1/81)cos(3x) |
Step 8: Combine the results.
Add up all the entries in the "∫v dx" column:
∫e^(2x)cos(3x)dx = (-1/6)e^(2x) - (1/3x + 2/27)e^(2x) + (1/27)sin(3x) - (1/81)cos(3x) + C
where C is the constant of integration.
That's how you evaluate the integral ∫e^(2x)cos(3x)dx using the tabular method.