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Chapter 5: Problem 102

Simplify the given expressions. $$\frac{d}{d x} \int_{x}^{1} e^{t^{2}} d t$$

Short Answer

Expert verified
Answer: The derivative of the integral is \(-e^{x^2}\).

Step by step solution

01

Apply the Fundamental Theorem of Calculus

Let's apply the Fundamental Theorem of Calculus to find the integral of \(e^{t^2}\) with limits from \(x\) to \(1\). Since the function is continuous, it has an antiderivative that we will denote as \(F(t)\). Then, the integral can be expressed as: $$\int_{x}^{1} e^{t^{2}} dt = F(1) - F(x)$$
02

Differentiate with respect to x

Now we need to find the derivative of the expression \(F(1) - F(x)\) with respect to \(x\). To do this, we'll differentiate both terms individually: $$\frac{d}{dx}(F(1) - F(x)) = \frac{d}{dx}(F(1)) - \frac{d}{dx}(F(x))$$ Since \(F(1)\) is a constant, its derivative with respect to \(x\) is zero. The derivative of \(F(x)\) with respect to \(x\) will be the function \(f(x)\), which in this case is \(e^{x^2}\), according to the Fundamental Theorem of Calculus: $$\frac{d}{dx}(F(1) - F(x)) = 0 - e^{x^2}$$
03

Final Answer

The derivative of the given integral with respect to \(x\) is: $$\frac{d}{d x} \int_{x}^{1} e^{t^{2}} d t = -e^{x^2}$$

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