FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD

�çe^3X DX

To find the indefinite integral of e^3x using the substitution method, we use the following steps:

Step 1: Choose a substitution.
We let u = 3x.

Step 2: Find the derivative of u with respect to x (du/dx).
Taking the derivative of both sides of the substitution equation gives us du/dx = 3.

Step 3: Solve for dx in terms of du.
Rearranging the equation gives us dx = du/3.

Step 4: Substitute the original function and dx in terms of du from steps 1 and 3 into the integral.
The original integral can now be rewritten as ∫ e^3x dx = ∫ e^u (du/3).

Step 5: Simplify the integral.
We can bring the (1/3) outside the integral sign to get (1/3) ∫ e^u du.

Step 6: Evaluate the new integral.
Integrating e^u with respect to u gives us ∫ e^u du = e^u + C, where C is the constant of integration.

Step 7: Substitute back the value of u.
Replacing u with 3x gives us e^(3x) + C, where C is the constant of integration.

Therefore, the indefinite integral of e^3x is e^(3x) + C.