Evaluate The Integral:$\[ \int \frac{2 \, Dt}{\left(t^2-4\right)^2} \\]

Introduction

In this article, we will evaluate the integral of a rational function, specifically the integral $\int \frac{2 \, dt}{\left(t^2-4\right)^2}$. This type of integral is a classic example of a rational function, and its evaluation requires a combination of algebraic and trigonometric techniques.

The Integral

The given integral is $\int \frac{2 \, dt}{\left(t^2-4\right)^2}$. To evaluate this integral, we can use the method of substitution. Let's substitute $u = t^2 - 4$, which implies $du = 2t \, dt$. We can rewrite the integral in terms of $u$ as follows:

$$\int \frac{2 \, dt}{\left(t^2-4\right)^2} = \int \frac{1}{u^2} \, du$$

Partial Fraction Decomposition

To evaluate the integral, we can use partial fraction decomposition. We can write the integral as:

$$\int \frac{1}{u^2} \, du = \int \left(\frac{A}{u} + \frac{B}{u^2}\right) \, du$$

where $A$ and $B$ are constants to be determined. Multiplying both sides by $u^2$, we get:

$$1 = A u + B$$

Solving for A and B

To solve for $A$ and $B$, we can equate the coefficients of like terms on both sides of the equation. We get:

$$A = 0$$

$$B = 1$$

Evaluating the Integral

Now that we have found the values of $A$ and $B$, we can evaluate the integral as follows:

$$\int \frac{1}{u^2} \, du = \int \left(\frac{0}{u} + \frac{1}{u^2}\right) \, du$$

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

$$= -\frac{1}{u} + C$$

Substituting Back

Now that we have evaluated the integral in terms of $u$, we can substitute back to get the final answer in terms of $t$. We get:

$$-\frac{1}{u} + C = -\frac{1}{t^2 - 4} + C$$

Conclusion

In this article, we evaluated the integral $\int \frac{2 \, dt}{\left(t^2-4\right)^2}$. We used the method of substitution to rewrite the integral in terms of a new variable $u$, and then used partial fraction decomposition to evaluate the integral. Finally, we substituted back to get the final answer in terms of $t$. The final answer is $\boxed{-\frac{1}{t^2 - 4} + C}$.

Additional Examples

Here are a few additional examples of evaluating integrals of rational functions:

• $\int \frac{1}{t^2 + 1} \, dt = \arctan t + C$
• $\int \frac{1}{t^2 - 1} \, dt = \frac{1}{2} \ln \left|\frac{t-1}{t+1}\right| + C$
• $\int \frac{1}{t^2 - 4t + 3} \, dt = \frac{1}{2} \ln \left|\frac{t-1}{t-3}\right| + C$

Tips and Tricks

Here are a few tips and tricks for evaluating integrals of rational functions:

• Use the method of substitution to rewrite the integral in terms of a new variable.
• Use partial fraction decomposition to evaluate the integral.
• Be careful when substituting back to get the final answer in terms of the original variable.
• Use the properties of logarithms and trigonometric functions to simplify the integral.

Conclusion

$$$$$$$$

Introduction

In our previous article, we evaluated the integral $\int \frac{2 \, dt}{\left(t^2-4\right)^2}$. We used the method of substitution to rewrite the integral in terms of a new variable $u$, and then used partial fraction decomposition to evaluate the integral. In this article, we will answer some common questions related to evaluating integrals of rational functions.

Q: What is the method of substitution?

A: The method of substitution is a technique used to evaluate integrals by substituting a new variable for a part of the integral. This can help simplify the integral and make it easier to evaluate.

Q: How do I choose the substitution?

A: When choosing a substitution, look for a part of the integral that can be written in the form of a derivative. For example, if the integral contains a term like $t^2 - 4$, you can substitute $u = t^2 - 4$.

Q: What is partial fraction decomposition?

A: Partial fraction decomposition is a technique used to break down a rational function into simpler fractions. This can help evaluate the integral by breaking it down into smaller, more manageable parts.

Q: How do I use partial fraction decomposition?

A: To use partial fraction decomposition, you need to write the rational function in the form of a sum of simpler fractions. For example, if the integral contains a term like $\frac{1}{t^2 - 4}$, you can write it as $\frac{A}{t - 2} + \frac{B}{t + 2}$.

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

A: Some common mistakes to avoid when evaluating integrals of rational functions include:

• Not using the correct substitution
• Not breaking down the rational function into simpler fractions
• Not simplifying the integral before evaluating it
• Not checking the final answer for errors

Q: How do I check my answer for errors?

A: To check your answer for errors, you can use the following steps:

• Plug the final answer back into the original integral
• Simplify the integral to see if it matches the original expression
• Check the final answer for any errors or inconsistencies

Q: What are some common integrals of rational functions?

A: Some common integrals of rational functions include:

• $\int \frac{1}{t^2 + 1} \, dt = \arctan t + C$
• $\int \frac{1}{t^2 - 1} \, dt = \frac{1}{2} \ln \left|\frac{t-1}{t+1}\right| + C$
• $\int \frac{1}{t^2 - 4t + 3} \, dt = \frac{1}{2} \ln \left|\frac{t-1}{t-3}\right| + C$

Q: How do I evaluate integrals of rational functions with repeated roots?

A: To evaluate integrals of rational functions with repeated roots, you can use the following steps:

• Factor the denominator to find the repeated root
• Use the method of substitution to rewrite the integral in terms of a new variable
• Use partial fraction decomposition to break down the rational function into simpler fractions
• Evaluate the integral using the simplified expression

Conclusion

In this article, we answered some common questions related to evaluating integrals of rational functions. We discussed the method of substitution, partial fraction decomposition, and common mistakes to avoid. We also provided some common integrals of rational functions and tips for evaluating integrals with repeated roots.