Evaluate the indefinite integral.
integral 2e^(2x)sin(e^2x)
Note: Use an upper-case "C" for the constant of integration.
first, let
u = e^(2x)
du = 2e^(2x) dx
∫2e^(2x)sin(e^(2x))dx
= ∫sin(u) du
I guess you can probably take it from there, no?
To evaluate this integral, you can use the technique of integration by parts. The integration by parts formula is given by:
∫ u dv = uv - ∫ v du
Let's assign u and dv to different parts of the integral:
u = sin(e^2x) (This will simplify after differentiating)
dv = 2e^(2x) dx (This will simplify after integrating)
To find du, we differentiate u with respect to x:
du = cos(e^2x) * d(e^2x) = cos(e^2x) * 2e^2x dx
To find v, we integrate dv with respect to x:
v = ∫ dv = ∫ 2e^(2x) dx = e^(2x)
Now we can substitute the values of u, v, du, and dv back into the integration by parts formula:
∫ 2e^(2x) sin(e^2x) dx = uv - ∫ v du
= sin(e^2x) * e^(2x) - ∫ e^(2x) * cos(e^2x) * 2e^2x dx
Simplifying this expression, we have:
∫ 2e^(2x) sin(e^2x) dx = sin(e^2x) * e^(2x) - 2e^2x * ∫ e^(2x) * cos(e^2x) dx
Now we have reduced the integral to another form. We still need to evaluate the remaining integral,
∫ e^(2x) * cos(e^2x) dx,
which can be done using integration by parts again.
By following these steps, we can evaluate the original given indefinite integral.