Integral of form e^ax sin(bx) dx or e^ax cos(bx) dx

To find the integral of the forms \( e^{ax} \sin(bx) \, dx \) or \( e^{ax} \cos(bx) \, dx \), we can use integration by parts or a specific formula derived from these techniques.

Integral of \( e^{ax} \sin(bx) \, dx \)

The integral can be computed using the formula:

Integral:

\[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C \]

Steps to Derive the Integral

• Apply integration by parts twice.
• Set up the equations for \( u \) and \( dv \).
• Combine the results to isolate the integral on one side.

Integral of \( e^{ax} \cos(bx) \, dx \)

Similarly, the integral for the cosine function is given by:

Integral:

\[ \int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \cos(bx) + b \sin(bx)) + C \]

Understanding the Components

In both cases, \( C \) represents the constant of integration. The terms \( a \) and \( b \) are constants that affect the amplitude and frequency of the sine and cosine functions.

These integrals are useful in various applications, including physics and engineering, particularly in solving differential equations.