How to integrate (e^x)*cos(x)?

To integrate (e^x)*cos(x), use integration by parts.

Integration by parts is a technique used to integrate the product of two functions. It involves choosing one function to differentiate and the other to integrate. In this case, let u = cos(x) and dv/dx = e^x. Then, du/dx = -sin(x) and v = e^x.

Using the formula for integration by parts, ∫u(dv/dx)dx = uv - ∫v(du/dx)dx, we have:

∫(e^x)*cos(x)dx = e^x*cos(x) - ∫(-sin(x)*e^x)dx

Now, we can integrate the remaining term using integration by parts again. Let u = -sin(x) and dv/dx = e^x. Then, du/dx = -cos(x) and v = e^x.

Using the formula for integration by parts again, we have:

∫(e^x)*cos(x)dx = e^x*cos(x) + e^x*sin(x) - ∫(e^x*(-cos(x)))dx

Simplifying the integral, we get:

∫(e^x)*cos(x)dx = e^x*cos(x) + e^x*sin(x) + ∫(e^x*cos(x))dx

Now, we have a new integral to solve. We can use integration by parts again, with u = cos(x) and dv/dx = e^x. Then, du/dx = -sin(x) and v = e^x.

Using the formula for integration by parts one more time, we have:

∫(e^x)*cos(x)dx = e^x*cos(x) + e^x*sin(x) + e^x*sin(x) - ∫(e^x*(-sin(x)))dx

Simplifying the integral, we get:

∫(e^x)*cos(x)dx = e^x*cos(x) + e^x*sin(x) + e^x*sin(x) - ∫(e^x*sin(x))dx

We now have a new integral to solve, but we can use integration by parts again with u = sin(x) and dv/dx = e^x. Then, du/dx = cos(x) and v = e^x.

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