Use The Table Of Integrals To Evaluate The Integral:\[$\int E^{5x} \sin(4x) \, Dx\$\]

Introduction

In this article, we will explore the use of the table of integrals to evaluate the integral of a product of exponential and trigonometric functions. The integral in question is $\int e^{5x} \sin(4x) \, dx$. We will use the properties of the table of integrals to simplify the integral and find its value.

The Table of Integrals

The table of integrals is a collection of known integrals that can be used to evaluate more complex integrals. It is a powerful tool in calculus that can be used to simplify and evaluate a wide range of integrals. The table of integrals includes integrals of trigonometric functions, exponential functions, logarithmic functions, and more.

Evaluating the Integral

To evaluate the integral $\int e^{5x} \sin(4x) \, dx$, we can use the table of integrals to simplify the integral. We can start by using the property of the table of integrals that states $\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx))$.

Applying the Property

We can apply the property of the table of integrals to the integral $\int e^{5x} \sin(4x) \, dx$. We have $a = 5$ and $b = 4$, so we can substitute these values into the property to get:

$$\int e^{5x} \sin(4x) \, dx = \frac{e^{5x}}{5^2 + 4^2} (5 \sin(4x) - 4 \cos(4x))$$

Simplifying the Integral

We can simplify the integral by evaluating the expression $\frac{e^{5x}}{5^2 + 4^2} (5 \sin(4x) - 4 \cos(4x))$. We have $5^2 + 4^2 = 25 + 16 = 41$, so we can substitute this value into the expression to get:

$$\int e^{5x} \sin(4x) \, dx = \frac{e^{5x}}{41} (5 \sin(4x) - 4 \cos(4x))$$

Final Answer

The final answer to the integral $\int e^{5x} \sin(4x) \, dx$ is $\frac{e^{5x}}{41} (5 \sin(4x) - 4 \cos(4x))$.

Conclusion

In this article, we used the table of integrals to evaluate the integral of a product of exponential and trigonometric functions. We applied the property of the table of integrals to simplify the integral and found its value. The table of integrals is a powerful tool in calculus that can be used to simplify and evaluate a wide range of integrals.

Common Integrals

Here are some common integrals that can be used to evaluate more complex integrals:

• $\int e^{ax} \, dx = \frac{e^{ax}}{a}$
• $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax)$
• $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax)$
• $\int \sin^2(ax) \, dx = \frac{1}{2a} (x - \frac{1}{2a} \sin(2ax))$
• $\int \cos^2(ax) \, dx = \frac{1}{2a} (x + \frac{1}{2a} \sin(2ax))$

Table of Integrals

Here is a table of integrals that can be used to evaluate more complex integrals:

Integral	Value
$\int e^{ax} \, dx$	$\frac{e^{ax}}{a}$
$\int \sin(ax) \, dx$	$-\frac{1}{a} \cos(ax)$
$\int \cos(ax) \, dx$	$\frac{1}{a} \sin(ax)$
$\int \sin^2(ax) \, dx$	$\frac{1}{2a} (x - \frac{1}{2a} \sin(2ax))$
$\int \cos^2(ax) \, dx$	$\frac{1}{2a} (x + \frac{1}{2a} \sin(2ax))$

References

• [1] "Calculus" by Michael Spivak
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "The Table of Integrals" by I.S. Gradshteyn and I.M. Ryzhik

Q: What is the table of integrals?

A: The table of integrals is a collection of known integrals that can be used to evaluate more complex integrals. It is a powerful tool in calculus that can be used to simplify and evaluate a wide range of integrals.

Q: How do I use the table of integrals to evaluate an integral?

A: To use the table of integrals to evaluate an integral, you need to identify the type of integral you are dealing with and match it to the corresponding entry in the table of integrals. Once you have matched the integral to the correct entry, you can use the formula provided in the table to evaluate the integral.

Q: What are some common integrals that can be used to evaluate more complex integrals?

A: Some common integrals that can be used to evaluate more complex integrals include:

• $\int e^{ax} \, dx = \frac{e^{ax}}{a}$
• $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax)$
• $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax)$
• $\int \sin^2(ax) \, dx = \frac{1}{2a} (x - \frac{1}{2a} \sin(2ax))$
• $\int \cos^2(ax) \, dx = \frac{1}{2a} (x + \frac{1}{2a} \sin(2ax))$

Q: How do I apply the property of the table of integrals to simplify an integral?

A: To apply the property of the table of integrals to simplify an integral, you need to identify the type of integral you are dealing with and match it to the corresponding entry in the table of integrals. Once you have matched the integral to the correct entry, you can use the formula provided in the table to simplify the integral.

Q: What are some examples of integrals that can be evaluated using the table of integrals?

A: Some examples of integrals that can be evaluated using the table of integrals include:

• $\int e^{5x} \sin(4x) \, dx = \frac{e^{5x}}{41} (5 \sin(4x) - 4 \cos(4x))$
• $\int \sin^2(2x) \, dx = \frac{1}{2} (x - \frac{1}{4} \sin(4x))$
• $\int \cos^2(3x) \, dx = \frac{1}{2} (x + \frac{1}{6} \sin(6x))$

Q: How do I use the table of integrals to evaluate a definite integral?

A: To use the table of integrals to evaluate a definite integral, you need to identify the type of integral you are dealing with and match it to the corresponding entry in the table of integrals. Once you have matched the integral to the correct entry, you can use the formula provided in the table to evaluate the definite integral.

Q: What are some common mistakes to avoid when using the table of integrals?

A: Some common mistakes to avoid when using the table of integrals include:

• Not matching the integral to the correct entry in the table of integrals
• Not using the correct formula to evaluate the integral
• Not checking the units of the integral
• Not checking the domain of the integral

Q: How do I find the table of integrals?

A: The table of integrals can be found in many calculus textbooks and online resources. Some popular online resources include:

• Wolfram Alpha
• Mathway
• Symbolab
• Khan Academy

Q: What are some advanced topics in calculus that involve the table of integrals?

A: Some advanced topics in calculus that involve the table of integrals include:

• Integration by parts
• Integration by partial fractions
• Integration by substitution
• Improper integrals
• Double integrals
• Triple integrals

Q: How do I use the table of integrals to evaluate a complex integral?

A: To use the table of integrals to evaluate a complex integral, you need to break down the integral into simpler components and match each component to the corresponding entry in the table of integrals. Once you have matched each component to the correct entry, you can use the formula provided in the table to evaluate the complex integral.

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

A: Some real-world applications of the table of integrals include:

• Physics: The table of integrals is used to evaluate integrals that appear in the study of physics, such as the integral of the force of gravity.
• Engineering: The table of integrals is used to evaluate integrals that appear in the study of engineering, such as the integral of the stress on a beam.
• Economics: The table of integrals is used to evaluate integrals that appear in the study of economics, such as the integral of the demand for a product.

Note: The questions and answers provided are for general information purposes only and may not be applicable to all situations. It is always best to consult a qualified professional or a reliable source for specific advice or guidance.