Finding the Original Function from its Derivative | Integrating 2xe^2 to Obtain y = x^2e^2 + C

y’=2xe²

The given expression, y’ = 2xe^2, represents the derivative of a function y with respect to x

The given expression, y’ = 2xe^2, represents the derivative of a function y with respect to x. To find the original function y, we need to integrate the given expression with respect to x.

To begin, let’s separate the variables dx and dy:

dy = 2xe^2 dx

Now, we can integrate both sides of the equation:

∫ dy = ∫ (2xe^2) dx

Integrating the left side with respect to y gives us:

y = ∫ (2xe^2) dx

To integrate the right side, we can use the power rule for integration and also take into account the exponential function.

∫ (2xe^2) dx = 2 ∫ (x e^2) dx

Applying the power rule, we integrate x to get (1/2)x^2, and treating e^2 as a constant, we get:

2 ∫ (x e^2) dx = 2 * (1/2)x^2 * e^2 + C

Where C is the constant of integration.

Simplifying further:

y = x^2 * e^2 + C

So, the original function y, that corresponds to the derivative y’ = 2xe^2, is y = x^2 * e^2 + C.

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