Use The Fundamental Theorem To Evaluate The Definite Integral Exactly.$\int_1^{25} \frac{3}{\sqrt{x}} \, Dx$Enter The Exact Answer.$\int_1^{25} \frac{3}{\sqrt{x}} \, Dx =$

Introduction

The Fundamental Theorem of Calculus (FTC) is a powerful tool for evaluating definite integrals. It states that if a function f(x) is continuous on the interval [a, b], then the definite integral of f(x) from a to b can be evaluated as the antiderivative of f(x) evaluated at b minus the antiderivative of f(x) evaluated at a. In this article, we will use the FTC to evaluate the definite integral $\int_1^{25} \frac{3}{\sqrt{x}} \, dx$ exactly.

Understanding the FTC

The FTC is a fundamental concept in calculus that relates the derivative of a function to the definite integral of that function. It states that if a function f(x) is continuous on the interval [a, b], then the definite integral of f(x) from a to b can be evaluated as:

$$\int_a^b f(x) \, dx = F(b) - F(a)$$

where F(x) is the antiderivative of f(x).

Evaluating the Definite Integral

To evaluate the definite integral $\int_1^{25} \frac{3}{\sqrt{x}} \, dx$, we need to find the antiderivative of the function $\frac{3}{\sqrt{x}}$. The antiderivative of $\frac{3}{\sqrt{x}}$ is $3 \cdot 2 \sqrt{x} = 6 \sqrt{x}$.

Applying the FTC

Now that we have found the antiderivative of the function, we can apply the FTC to evaluate the definite integral. We have:

$$\int_1^{25} \frac{3}{\sqrt{x}} \, dx = 6 \sqrt{x} \Big|_1^{25}$$

Evaluating the antiderivative at the upper and lower limits of integration, we get:

$$6 \sqrt{x} \Big|_1^{25} = 6 \sqrt{25} - 6 \sqrt{1}$$

Simplifying, we get:

$$6 \sqrt{25} - 6 \sqrt{1} = 6 \cdot 5 - 6 \cdot 1 = 30 - 6 = 24$$

Therefore, the exact value of the definite integral $\int_1^{25} \frac{3}{\sqrt{x}} \, dx$ is 24.

Conclusion

In this article, we used the Fundamental Theorem of Calculus to evaluate the definite integral $\int_1^{25} \frac{3}{\sqrt{x}} \, dx$ exactly. We found the antiderivative of the function $\frac{3}{\sqrt{x}}$ and applied the FTC to evaluate the definite integral. The exact value of the definite integral is 24.

Additional Examples

Here are a few additional examples of evaluating definite integrals using the FTC:

• $\int_0^2 x^2 \, dx = \frac{x^3}{3} \Big|_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$
• $\int_1^3 \frac{1}{x} \, dx = \ln|x| \Big|_1^3 = \ln|3| - \ln|1| = \ln 3$
• $\int_0^1 e^x \, dx = e^x \Big|_0^1 = e^1 - e^0 = e - 1$

These examples demonstrate the power of the FTC in evaluating definite integrals exactly.

Final Thoughts

Introduction

In our previous article, we used the Fundamental Theorem of Calculus (FTC) to evaluate the definite integral $\int_1^{25} \frac{3}{\sqrt{x}} \, dx$ exactly. In this article, we will answer some common questions related to evaluating definite integrals using the FTC.

Q: What is the Fundamental Theorem of Calculus?

A: The Fundamental Theorem of Calculus is a fundamental concept in calculus that relates the derivative of a function to the definite integral of that function. It states that if a function f(x) is continuous on the interval [a, b], then the definite integral of f(x) from a to b can be evaluated as the antiderivative of f(x) evaluated at b minus the antiderivative of f(x) evaluated at a.

Q: How do I find the antiderivative of a function?

A: To find the antiderivative of a function, you need to find a function whose derivative is the original function. For example, the antiderivative of x^2 is (x^3)/3, and the antiderivative of e^x is e^x.

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

A: The FTC and the power rule are both used to evaluate definite integrals, but they are used in different ways. The power rule is used to evaluate definite integrals of the form $\int x^n \, dx$, while the FTC is used to evaluate definite integrals of the form $\int f(x) \, dx$ where f(x) is a continuous function.

Q: Can I use the FTC to evaluate definite integrals of trigonometric functions?

A: Yes, you can use the FTC to evaluate definite integrals of trigonometric functions. For example, the antiderivative of sin(x) is -cos(x), and the antiderivative of cos(x) is sin(x).

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

A: The FTC and the chain rule are both used to evaluate definite integrals, but they are used in different ways. The chain rule is used to evaluate definite integrals of the form $\int f(g(x)) \, dx$, while the FTC is used to evaluate definite integrals of the form $\int f(x) \, dx$ where f(x) is a continuous function.

Q: Can I use the FTC to evaluate definite integrals of rational functions?

A: Yes, you can use the FTC to evaluate definite integrals of rational functions. For example, the antiderivative of 1/x is ln|x|, and the antiderivative of x^2 is (x^3)/3.

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

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

• Not checking if the function is continuous on the interval [a, b]
• Not finding the antiderivative of the function
• Not evaluating the antiderivative at the upper and lower limits of integration
• Not simplifying the expression

Conclusion

In this article, we answered some common questions related to evaluating definite integrals using the Fundamental Theorem of Calculus. We discussed the FTC, antiderivatives, and common mistakes to avoid when using the FTC. We hope this article has been helpful in understanding the FTC and how to use it to evaluate definite integrals.

Additional Resources

For more information on the FTC and how to use it to evaluate definite integrals, we recommend the following resources:

• Calculus textbooks, such as "Calculus" by Michael Spivak or "Calculus" by James Stewart
• Online resources, such as Khan Academy or MIT OpenCourseWare
• Calculus courses, such as Calculus I, II, or III

We hope this article has been helpful in understanding the FTC and how to use it to evaluate definite integrals. If you have any further questions, please don't hesitate to ask.