Evaluate The Derivative Of The Integral:${ \frac{d}{d X} \int_{3 X 2} 1 \frac{2}{1+u^3} , Du }$

Mar 10, 2025 by ADMIN 97 views

Evaluate the Derivative of the Integral

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Introduction

In calculus, the Fundamental Theorem of Calculus (FTC) establishes a deep connection between differentiation and integration. The FTC states that differentiation and integration are inverse processes, and this relationship is crucial in solving problems involving definite integrals. In this article, we will evaluate the derivative of the integral $\frac{d}{d x} \int_{3 x^2}^1 \frac{2}{1+u^3} \, du$ using the FTC.

The Fundamental Theorem of Calculus

The FTC consists of two parts: the first part relates the derivative of an integral to the original function, while the second part relates the definite integral to the antiderivative of the function. The first part of the FTC states that if $f(x)$ is a continuous function on the interval $[a, b]$ , then the derivative of the definite integral of $f(x)$ with respect to $x$ is equal to $f(x)$ .

Mathematically, this can be expressed as:

\frac{d}{dx} \int_{a}^{b} f(x) \, dx = f(x)

Applying the Fundamental Theorem of Calculus

To evaluate the derivative of the integral $\frac{d}{d x} \int_{3 x^2}^1 \frac{2}{1+u^3} \, du$ , we can use the first part of the FTC. According to the FTC, the derivative of the integral is equal to the original function evaluated at the upper limit of integration.

In this case, the original function is $\frac{2}{1+u^3}$ , and the upper limit of integration is $1$ . Therefore, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{3 x^2}^1 \frac{2}{1+u^3} \, du = \frac{2}{1+(1)^3}

Simplifying the Expression

We can simplify the expression $\frac{2}{1+(1)^3}$ by evaluating the exponent and the denominator.

Since $(1)^3 = 1$ , the denominator becomes $1 + 1 = 2$ . Therefore, the expression simplifies to:

\frac{2}{2} = 1

Conclusion

In this article, we evaluated the derivative of the integral $\frac{d}{d x} \int_{3 x^2}^1 \frac{2}{1+u^3} \, du$ using the Fundamental Theorem of Calculus. We applied the first part of the FTC to find that the derivative of the integral is equal to the original function evaluated at the upper limit of integration. Finally, we simplified the expression to obtain the final answer.

Example Problems

Problem 1

Evaluate the derivative of the integral $\frac{d}{d x} \int_{x^2}^1 \frac{3}{1+u^2} \, du$ .

Solution

Using the first part of the FTC, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{x^2}^1 \frac{3}{1+u^2} \, du = \frac{3}{1+(x^2)^2}

Problem 2

Evaluate the derivative of the integral $\frac{d}{d x} \int_{2x}^1 \frac{4}{1+u^4} \, du$ .

Solution

Using the first part of the FTC, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{2x}^1 \frac{4}{1+u^4} \, du = \frac{4}{1+(2x)^4}

Final Answer

The final answer is $\boxed{1}$ .

Step-by-Step Solution

Step 1: Evaluate the upper limit of integration

The upper limit of integration is $1$ .

Step 2: Evaluate the original function at the upper limit of integration

The original function is $\frac{2}{1+u^3}$ , and the upper limit of integration is $1$ . Therefore, we can evaluate the original function as follows:

\frac{2}{1+(1)^3} = \frac{2}{2} = 1

Step 3: Simplify the expression

The expression $\frac{2}{2}$ simplifies to $1$ .

Step 4: Evaluate the derivative of the integral

Using the first part of the FTC, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{3 x^2}^1 \frac{2}{1+u^3} \, du = \frac{2}{1+(1)^3} = 1

Final Answer

The final answer is $\boxed{1}$ .

Step-by-Step Solution

Step 1: Evaluate the upper limit of integration

The upper limit of integration is $1$ .

Step 2: Evaluate the original function at the upper limit of integration

The original function is $\frac{2}{1+u^3}$ , and the upper limit of integration is $1$ . Therefore, we can evaluate the original function as follows:

\frac{2}{1+(1)^3} = \frac{2}{2} = 1

Step 3: Simplify the expression

The expression $\frac{2}{2}$ simplifies to $1$ .

Step 4: Evaluate the derivative of the integral

Using the first part of the FTC, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{3 x^2}^1 \frac{2}{1+u^3} \, du = \frac{2}{1+(1)^3} = 1

Final Answer

The final answer is $\boxed{1}$ .

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Frequently Asked Questions

Q: What is the Fundamental Theorem of Calculus (FTC)?

A: The Fundamental Theorem of Calculus (FTC) is a theorem in calculus that establishes a deep connection between differentiation and integration. It states that differentiation and integration are inverse processes.

Q: What is the first part of the FTC?

A: The first part of the FTC states that if $f(x)$ is a continuous function on the interval $[a, b]$ , then the derivative of the definite integral of $f(x)$ with respect to $x$ is equal to $f(x)$ .

Q: How do I apply the FTC to evaluate the derivative of an integral?

A: To apply the FTC, you need to identify the original function and the upper limit of integration. Then, you can evaluate the derivative of the integral by substituting the upper limit of integration into the original function.

Q: What is the difference between the FTC and the chain rule?

A: The FTC and the chain rule are both used to evaluate the derivative of a composite function. However, the FTC is used to evaluate the derivative of a definite integral, while the chain rule is used to evaluate the derivative of a composite function.

Q: Can I use the FTC to evaluate the derivative of an indefinite integral?

A: No, the FTC is only used to evaluate the derivative of a definite integral. To evaluate the derivative of an indefinite integral, you need to use the chain rule.

Q: What are some common mistakes to avoid when applying the FTC?

A: Some common mistakes to avoid when applying the FTC include:

Failing to identify the original function and the upper limit of integration
Failing to substitute the upper limit of integration into the original function
Failing to simplify the expression after substituting the upper limit of integration

Q: Can I use the FTC to evaluate the derivative of a function that is not continuous?

A: No, the FTC requires that the function be continuous on the interval $[a, b]$ . If the function is not continuous, you cannot use the FTC to evaluate the derivative of the integral.

Q: What are some real-world applications of the FTC?

A: The FTC has many real-world applications, including:

Physics: The FTC is used to evaluate the work done by a force on an object
Engineering: The FTC is used to evaluate the energy required to perform a task
Economics: The FTC is used to evaluate the marginal cost of a product

Example Problems

Problem 1

Evaluate the derivative of the integral $\frac{d}{d x} \int_{x^2}^1 \frac{3}{1+u^2} \, du$ .

Solution

Using the first part of the FTC, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{x^2}^1 \frac{3}{1+u^2} \, du = \frac{3}{1+(x^2)^2}

Problem 2

Evaluate the derivative of the integral $\frac{d}{d x} \int_{2x}^1 \frac{4}{1+u^4} \, du$ .

Solution

Using the first part of the FTC, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{2x}^1 \frac{4}{1+u^4} \, du = \frac{4}{1+(2x)^4}

Final Answer

The final answer is $\boxed{1}$ .

Step-by-Step Solution

Step 1: Evaluate the upper limit of integration

The upper limit of integration is $1$ .

Step 2: Evaluate the original function at the upper limit of integration

The original function is $\frac{2}{1+u^3}$ , and the upper limit of integration is $1$ . Therefore, we can evaluate the original function as follows:

\frac{2}{1+(1)^3} = \frac{2}{2} = 1

Step 3: Simplify the expression

The expression $\frac{2}{2}$ simplifies to $1$ .

Step 4: Evaluate the derivative of the integral

Using the first part of the FTC, we can evaluate the derivative of the integral as follows:

\frac{d}{dx} \int_{3 x^2}^1 \frac{2}{1+u^3} \, du = \frac{2}{1+(1)^3} = 1

Final Answer

The final answer is $\boxed{1}$ .