FIND EACH INDEFINITE INTERGRAL BY THE SUBSTITUTION METHOD
e^3X DX
To find the indefinite integral of e^(3x) dx using the substitution method, you can follow these steps:
Step 1: Let u = 3x.
Differentiating both sides with respect to x will give you du = 3 dx (by the chain rule).
Step 2: Rearrange the equation from Step 1 to solve for dx:
dx = du/3.
Step 3: Substitute the expression for dx from Step 2 and the expression for u from Step 1 into the original integral:
∫ e^(3x) dx = ∫ e^u * (du/3).
Step 4: Simplify the integral:
∫ e^u * (du/3) = (1/3) ∫ e^u du.
Step 5: Integrate the simplified integral:
∫ e^u du = e^u + C.
Step 6: Substitute the expression for u back into the solution from Step 5:
e^u + C = e^(3x) + C.
Therefore, the indefinite integral of e^(3x) dx using the substitution method is e^(3x) + C, where C is the constant of integration.