integrate 2^(3x). 3^(2x) . 5^(x) dx
∫a^(cx) dx = 1/(c lna) a^x
now just use the product rule
oops. how could I have mixed up derivatives with integrals? There is no product rule.
However, we do have 8^x * 9^x * 5^x = 360^x
which should be easy to do, no?
To integrate the expression ∫2^(3x) * 3^(2x) * 5^(x) dx, we can use the properties of exponents and the rules of integration. Here's the step-by-step process:
Step 1: Rewrite the expression using the properties of exponents.
∫(2^3)^x * (3^2)^x * 5^x dx
Step 2: Simplify the expression.
∫8^x * 9^x * 5^x dx
Step 3: Combine the bases.
∫(8 * 9 * 5)^x dx
Step 4: Evaluate the constant.
∫360^x dx
Step 5: Apply the power rule of integration.
= (1/(ln(360))) * (360^x) + C
So, the integral of 2^(3x) * 3^(2x) * 5^(x) dx is (1/(ln(360))) * (360^x) + C, where C is the constant of integration.