Question

Use the Tabular method: Evaluate int e^{-3 x} sin (2 x) d x

Answer 1

To evaluate the given integral, we can use integration by parts. By defining u = sin(2x) and dv = e^-3x dx, we can apply the tabular method and set up a table to find du and v. Using the integration by parts formula, we can then plug in the values from the table and simplify the integral.

Step-by-step explanation:

To evaluate the integral ∫ e-3xsin(2x) dx using the tabular method, we need to use integration by parts. Let's define u = sin(2x) and dv = e-3x dx. Then, we can find du and v by differentiating and integrating respectively.

Using the tabular method, we can set up the table as follows:

udvduvsin(2x)e-3x dx2cos(2x) dx-1/3e-3x

Next, we can apply the integration by parts formula:

∫ u dv = uv - ∫ v du

Plugging in the values from the table:

∫ e-3xsin(2x) dx = -1/3e-3xsin(2x) - ∫ -1/3e-3x2cos(2x) dx

Simplifying the integral on the right-hand side, we can further solve it using the tabular method or directly integrate.

Remember to evaluate the definite integral if a specific interval is given.