What is the antiderivative of. 3X/x^2?
(c) find the integral of e3x + ex dx
3x/x^2 = 3/x
so, 3 ln(x)
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1/3 e^3x + e^x
To find the antiderivative of the function (3x/x^2), we can follow these steps:
Step 1: Separate the function into two parts. In this case, 3x can be written as (3 * x^1) and x^2 can be written as x^(2-1).
Step 2: Apply the power rule of integration, where the antiderivative of x^n is (x^(n+1))/(n+1).
Using the power rule, we can integrate each part separately:
For (3 * x^1), we add 1 to the power of x and divide by the new power:
∫(3 * x^1) dx = (3/2) * x^2 + C
For x^(2-1), we add 1 to the power of x and divide by the new power:
∫x^(2-1) dx = (1/2) * x^2 + C
Step 3: Combine the results from each part:
∫(3x/x^2) dx = (3/2) * x^2 + (1/2) * x^2 + C = (4/2) * x^2 + C = 2x^2 + C
Therefore, the antiderivative of 3x/x^2 is 2x^2 + C, where C is the constant of integration.