What is the value of the integral of e^(ax) * cos(bx) dx?

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The integral of the function eax cos(bx) can be solved using integration by parts or through the method of complex numbers. However, one of the most effective ways to address this integral is by recognizing it as a standard form.

To evaluate the integral, we can express it as:

I = ∫ eax cos(bx) dx

Using the integral of the exponential function combined with trigonometric functions, we can apply the following result:

∫ eax cos(bx) dx = eax × (A cos(bx) + B sin(bx)) / (a2 + b2) + C

where:

  • A = a and
  • B = b
  • C is the constant of integration.

Putting it all together, we can summarize that:

∫ eax cos(bx) dx = eax × (a cos(bx) + b sin(bx)) / (a2 + b2) + C

This result showcases how the integral results in a function involving both sine and cosine, multiplied by the exponential. It’s an elegant solution to a common integral that often appears in both engineering and mathematical contexts.

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