Evaluate The Integral:$\[ \int \frac{1}{x^2+4x+5} \, Dx \\]

Feb 26, 2025 by ADMIN 60 views

**Evaluating the Integral of a Rational Function**

Introduction

In this article, we will delve into the evaluation of a specific integral, which is a fundamental concept in calculus. The integral in question is $\int \frac{1}{x^2+4x+5} \, dx$ . This type of integral is known as a rational function, and it can be challenging to evaluate. However, with the right techniques and strategies, we can break down the integral into manageable parts and find its solution.

Understanding the Integral

The given integral is a rational function, which means it is a ratio of two polynomials. In this case, the numerator is a constant (1), and the denominator is a quadratic polynomial ( $x^2+4x+5$ ). To evaluate this integral, we need to find a way to simplify the denominator and then use substitution or other techniques to integrate the resulting expression.

Simplifying the Denominator

One way to simplify the denominator is to complete the square. This involves rewriting the quadratic polynomial in the form $(x+a)^2+b$ . In this case, we can rewrite the denominator as follows:

$x^2+4x+5 = (x+2)^2+1$

This simplification allows us to rewrite the integral as:

$\int \frac{1}{(x+2)^2+1} \, dx$

Using Substitution

Now that we have simplified the denominator, we can use substitution to evaluate the integral. Let's substitute $u = x+2$ . This means that $du = dx$ , and we can rewrite the integral as:

$\int \frac{1}{u^2+1} \, du$

Evaluating the Integral

The integral $\int \frac{1}{u^2+1} \, du$ is a standard integral, and its solution is well-known. The antiderivative of $\frac{1}{u^2+1}$ is $\arctan(u)$ . Therefore, we can evaluate the integral as follows:

$\int \frac{1}{u^2+1} \, du = \arctan(u) + C$

Substituting Back

Now that we have evaluated the integral, we can substitute back to find the solution in terms of $x$ . Recall that $u = x+2$ , so we can substitute this expression back into the solution:

$\arctan(u) + C = \arctan(x+2) + C$

Conclusion

In this article, we evaluated the integral $\int \frac{1}{x^2+4x+5} \, dx$ using the technique of substitution. We simplified the denominator by completing the square, and then used substitution to evaluate the resulting integral. The solution to the integral is $\arctan(x+2) + C$ , where $C$ is the constant of integration.

Applications of the Integral

The integral $\int \frac{1}{x^2+4x+5} \, dx$ has several applications in mathematics and physics. For example, it can be used to model the motion of an object under the influence of a force that is proportional to the distance from a fixed point. The integral can also be used to solve problems involving electrical circuits and optics.

Future Directions

In this article, we evaluated a specific integral using the technique of substitution. However, there are many other techniques and strategies that can be used to evaluate integrals. Some of these techniques include integration by parts, integration by partial fractions, and the use of trigonometric substitution. In future articles, we can explore these techniques in more detail and apply them to a wide range of problems.

References

[1] "Calculus" by Michael Spivak
[2] "Introduction to Calculus" by Michael Sullivan
[3] "Calculus: Early Transcendentals" by James Stewart

Glossary

Rational function: A ratio of two polynomials.
Quadratic polynomial: A polynomial of degree two.
Substitution: A technique used to evaluate integrals by replacing a variable with a new expression.
Antiderivative: A function that is the derivative of another function.
Constant of integration: A constant that is added to the solution of an integral to make it exact.

Practice Problems

To practice evaluating integrals, try the following problems:

Evaluate the integral $\int \frac{1}{x^2+2x+2} \, dx$ .
Evaluate the integral $\int \frac{1}{x^2-4x+3} \, dx$ .
Evaluate the integral $\int \frac{1}{x^2+6x+8} \, dx$ .

Conclusion

Introduction

In our previous article, we evaluated the integral $\int \frac{1}{x^2+4x+5} \, dx$ using the technique of substitution. We simplified the denominator by completing the square, and then used substitution to evaluate the resulting integral. In this article, we will answer some common questions that readers may have about the integral and its evaluation.

Q: What is the significance of completing the square in the denominator?

A: Completing the square in the denominator allows us to rewrite the quadratic polynomial in a form that is easier to integrate. By rewriting the denominator as $(x+2)^2+1$ , we can use substitution to evaluate the integral.

Q: Why did we use substitution to evaluate the integral?

A: We used substitution to evaluate the integral because it allowed us to simplify the denominator and make the integral easier to integrate. By substituting $u = x+2$ , we were able to rewrite the integral in a form that was more manageable.

Q: What is the antiderivative of $\frac{1}{u^2+1}$ ?

A: The antiderivative of $\frac{1}{u^2+1}$ is $\arctan(u)$ . This is a standard integral that can be evaluated using the substitution $u = \tan(x)$ .

Q: How do we substitute back to find the solution in terms of $x$ ?

A: To substitute back, we need to replace $u$ with $x+2$ in the solution. This gives us $\arctan(x+2) + C$ , where $C$ is the constant of integration.

Q: What are some common applications of the integral $\int \frac{1}{x^2+4x+5} \, dx$ ?

A: The integral $\int \frac{1}{x^2+4x+5} \, dx$ has several applications in mathematics and physics. For example, it can be used to model the motion of an object under the influence of a force that is proportional to the distance from a fixed point. The integral can also be used to solve problems involving electrical circuits and optics.

Q: What are some other techniques that can be used to evaluate integrals?

A: There are several other techniques that can be used to evaluate integrals, including integration by parts, integration by partial fractions, and the use of trigonometric substitution. These techniques can be used to evaluate a wide range of integrals, including those that involve rational functions, trigonometric functions, and exponential functions.

Q: How can I practice evaluating integrals?

A: There are several ways to practice evaluating integrals, including working through practice problems and using online resources such as Khan Academy and Wolfram Alpha. You can also try solving problems from calculus textbooks or online resources.

Q: What are some common mistakes to avoid when evaluating integrals?

A: Some common mistakes to avoid when evaluating integrals include:

Not simplifying the denominator before evaluating the integral
Not using substitution or other techniques to simplify the integral
Not checking the solution for accuracy
Not considering the constant of integration

Conclusion

In this article, we answered some common questions that readers may have about the integral $\int \frac{1}{x^2+4x+5} \, dx$ and its evaluation. We discussed the significance of completing the square in the denominator, the use of substitution to evaluate the integral, and some common applications of the integral. We also provided some practice problems and tips for avoiding common mistakes when evaluating integrals.

Practice Problems

To practice evaluating integrals, try the following problems:

Evaluate the integral $\int \frac{1}{x^2+2x+2} \, dx$ .
Evaluate the integral $\int \frac{1}{x^2-4x+3} \, dx$ .
Evaluate the integral $\int \frac{1}{x^2+6x+8} \, dx$ .

References

[1] "Calculus" by Michael Spivak
[2] "Introduction to Calculus" by Michael Sullivan
[3] "Calculus: Early Transcendentals" by James Stewart

Glossary

Rational function: A ratio of two polynomials.
Quadratic polynomial: A polynomial of degree two.
Substitution: A technique used to evaluate integrals by replacing a variable with a new expression.
Antiderivative: A function that is the derivative of another function.
Constant of integration: A constant that is added to the solution of an integral to make it exact.

Evaluate The Integral:$\[ \int \frac{1}{x^2+4x+5} \, Dx \\]

Introduction

Understanding the Integral

Simplifying the Denominator

Using Substitution

Evaluating the Integral

Substituting Back

Conclusion

Applications of the Integral

Future Directions

References

Glossary

Further Reading

Practice Problems

Conclusion

Introduction

Q: What is the significance of completing the square in the denominator?

Q: Why did we use substitution to evaluate the integral?

Q: What is the antiderivative of $\frac{1}{u^2+1}$ ?

Q: How do we substitute back to find the solution in terms of $x$ ?

Q: What are some common applications of the integral $\int \frac{1}{x^2+4x+5} \, dx$ ?

Q: What are some other techniques that can be used to evaluate integrals?

Q: How can I practice evaluating integrals?

Q: What are some common mistakes to avoid when evaluating integrals?

Conclusion

Practice Problems

References

Glossary

Further Reading

Introduction

Understanding the Integral

Simplifying the Denominator

Using Substitution

Evaluating the Integral

Substituting Back

Conclusion

Applications of the Integral

Future Directions

References

Glossary

Further Reading

Practice Problems

Conclusion

Introduction

Q: What is the significance of completing the square in the denominator?

Q: Why did we use substitution to evaluate the integral?

Q: What is the antiderivative of 1u2+1\frac{1}{u^2+1}u2+11​?

Q: How do we substitute back to find the solution in terms of xxx?

Q: What are some common applications of the integral ∫1x2+4x+5 dx\int \frac{1}{x^2+4x+5} \, dx∫x2+4x+51​dx?

Q: What are some other techniques that can be used to evaluate integrals?

Q: How can I practice evaluating integrals?

Q: What are some common mistakes to avoid when evaluating integrals?

Conclusion

Practice Problems

References

Glossary

Further Reading

Q: What is the antiderivative of $\frac{1}{u^2+1}$ ?

Q: How do we substitute back to find the solution in terms of $x$ ?

Q: What are some common applications of the integral $\int \frac{1}{x^2+4x+5} \, dx$ ?