Question

In Exercises \(7-12,\) use differentiation to verify the antiderivative formula. $$\int e^{2 x} d x=\frac{1}{2} e^{2 x}+C$$

Short answer

The thorough differentiation of the provided antiderivative \(\frac{1}{2} e^{2 x}+C\) was computed and confirmed correctly to equal \(e^{2x}\), validating the original antiderivative formula.

Step-by-step solution

Understanding the Concept

Differentiation and integration are two fundamental operations in calculus that are essentially inverse operations of each other. Thus, deriving the right-hand side of the antiderivative formula should give us the integrand on the left-hand side.

Differentiate the Antiderivative

First, compute the derivative of \(\frac{1}{2} e^{2 x}+C\) using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function, times the derivative of the inner function. In this case, the outer function is \(e^x\) and the inner function is \(2x\).

Compute the Derivative

The derivative of \(\frac{1}{2} e^{2 x}+C\) is \(e^{2x}\) which is in line with the integrand on the left-hand side of the formula. The constant \(C\) disappears when taking the derivative, because the derivative of a constant is zero.

Confirming the Antiderivative Formula

The derivative of \(\frac{1}{2} e^{2 x}+C\) is \(e^{2x}\), which matches the integrand in the given formula, therefore, confirming the antiderivative formula.