What is the integral of (e^x) cos(x) dx?

What is the integral of (e^x) cos(x) dx?

• A. (1/2)e^x (sin(x) + cos(x)) + C
• B. e^x sin(x) + C
• C. (1/2)e^x sin(2x) + C
• D. e^x (cos(x) - sin(x)) + C

Answer: (A)

To determine the integral of \( e^x \cos(x) \, dx \), we can utilize the method of integration by parts or employ the technique known as integration by parts multiple times. Starting with the integral \( I = \int e^x \cos(x) \, dx \), we can set it up for integration by parts. In integration by parts, we let: - \( u = \cos(x) \) and \( dv = e^x \, dx \) This leads to: - \( du = -\sin(x) \, dx \) - \( v = e^x \) Applying integration by parts, we get: \[ I = e^x \cos(x) - \int e^x (-\sin(x)) \, dx \] This simplifies to: \[ I = e^x \cos(x) + \int e^x \sin(x) \, dx \] Now, let’s define \( J = \int e^x \sin(x) \, dx \). We can again apply integration by parts to \( J \). Choosing: - \( u = \sin(x) \) and \( dv = e^x \,

To determine the integral of ( e^x \cos(x) , dx ), we can utilize the method of integration by parts or employ the technique known as integration by parts multiple times.

Starting with the integral ( I = \int e^x \cos(x) , dx ), we can set it up for integration by parts. In integration by parts, we let:

• ( u = \cos(x) ) and ( dv = e^x , dx )

This leads to:

• ( du = -\sin(x) , dx )

• ( v = e^x )

Applying integration by parts, we get:

[

I = e^x \cos(x) - \int e^x (-\sin(x)) , dx

]

This simplifies to:

[

I = e^x \cos(x) + \int e^x \sin(x) , dx

]

Now, let’s define ( J = \int e^x \sin(x) , dx ). We can again apply integration by parts to ( J ).

Choosing:

• ( u = \sin(x) ) and ( dv = e^x ,