Final answer:
To find the antiderivative of ∫cos(2x)eˣdx, you can use integration by parts. Assign u = cos(2x) and dv = eˣ dx, then apply the integration by parts formula. By repeating the process, you can solve for the original integral.
Step-by-step explanation:
To find the antiderivative of ∫cos(2x)eˣdx, we can use integration by parts. Integration by parts is based on the formula ∫u dv = uv - ∫v du. Let's assign u = cos(2x) and dv = eˣ dx. Taking the derivatives of u and the antiderivative of dv, we get du = -2sin(2x) dx and v = eˣ. Substituting these values into the integration by parts formula, we have:
∫cos(2x)eˣdx = cos(2x)eˣ - ∫(-2sin(2x)eˣ)dx = cos(2x)eˣ + 2∫sin(2x)eˣdx
We now have another integral to solve, but this time we can use integration by parts again. Let u = sin(2x) and dv = eˣ dx. Taking the derivatives of u and the antiderivative of dv, we get du = 2cos(2x) dx and v = eˣ. Substituting these values into the formula, we have:
∫sin(2x)eˣdx = sin(2x)eˣ - ∫(2cos(2x)eˣ)dx = sin(2x)eˣ - 2∫cos(2x)eˣdx
At this point, we can rearrange the equation to solve for the original integral:
3∫cos(2x)eˣdx = cos(2x)eˣ + 2∫sin(2x)eˣdx
3∫cos(2x)eˣdx + 2∫cos(2x)eˣdx = cos(2x)eˣ + ∫sin(2x)eˣdx
5∫cos(2x)eˣdx = cos(2x)eˣ + ∫sin(2x)eˣdx
Dividing both sides by 5, we get:
∫cos(2x)eˣdx = (cos(2x)eˣ + ∫sin(2x)eˣdx) / 5