Integrate e^cos 3x dx
There is no elementary antiderivative of e^cos 3x dx.
To integrate e^cos(3x) dx, we can use the substitution method.
Let's start by making the substitution:
Let u = cos(3x).
Differentiating both sides with respect to x:
du/dx = -3sin(3x)
Dividing both sides by -3:
(-1/3) du = sin(3x) dx
dx = (-3/3) du = -du
Now, we can rewrite the integral in terms of u:
∫ e^cos(3x) dx = ∫ e^u (-du)
Next, we can integrate with respect to u:
∫ e^u (-du) = - ∫ e^u du
The integral of e^u is simply e^u:
- ∫ e^u du = -e^u + C
Finally, substituting u back in terms of x:
- e^cos(3x) + C
Therefore, the final antiderivative of e^cos(3x) is -e^cos(3x) + C, where C is the constant of integration.