To solve the integral ∫ e^{ax} cos(bx) dx, we can use integration by parts or consider the method of complex exponentials. Here’s a step-by-step breakdown:
First, we recognize that the integral can be tackled using the formula for the integral of a product of an exponential function and a trigonometric function. The result can be derived using the technique of integration by parts or using the formula:
Let:
- u = e^{ax}
- dv = cos(bx) dx
Then, we compute:
- du = a e^{ax} dx
- v = (1/b) sin(bx)
Using integration by parts:
∫ u dv = uv - ∫ v du
Applying this to our integral, we have:
∫ e^{ax} cos(bx) dx = e^{ax} * (1/b) sin(bx) - ∫ (1/b) sin(bx) * a e^{ax} dxRearranging gives us:
∫ e^{ax} cos(bx) dx = (e^{ax} / b) sin(bx) - (a/b) ∫ e^{ax} sin(bx) dxThe integral ∫ e^{ax} sin(bx) dx can be solved similarly using integration by parts. By solving both integrals together, you’ll eventually arrive at a solution that combines both results.
The final result for the integral can be expressed as:
∫ e^{ax} cos(bx) dx = (e^{ax} / (a^2 + b^2)) (a cos(bx) + b sin(bx)) + CWhere C is the integration constant. This formula gives you the value of the integral for any constants a and b.