The Integral ∫ X X 2 + 1 D X \int \frac{x}{x^2+1} \, Dx ∫ X 2 + 1 X ​ D X Can Be Solved Using:A. Substitution B. Partial Fractions C. Integration By Parts D. None Of The Above

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Introduction

When it comes to solving integrals, there are several methods that can be employed, depending on the complexity of the function. In this case, we are tasked with finding the integral of $\frac{x}{x^2+1}$. This is a classic example of an integral that can be solved using a variety of techniques, including substitution, partial fractions, and integration by parts. In this article, we will explore each of these methods and determine which one is the most suitable for solving this particular integral.

Method A: Substitution

One of the most common methods for solving integrals is substitution. This involves replacing a portion of the integral with a new variable, which can simplify the expression and make it easier to integrate. In the case of the integral $\int \frac{x}{x^2+1} \, dx$, we can use substitution to rewrite the integral in a more manageable form.

Let's start by defining a new variable, $u = x^2 + 1$. This means that $du = 2x \, dx$, or $x \, dx = \frac{1}{2} du$. We can now substitute these expressions into the original integral:

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

This simplifies the integral to:

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

Now, we can integrate this expression with respect to $u$:

$$\int \frac{1}{2u} \, du = \frac{1}{2} \ln |u| + C$$

Substituting back in for $u = x^2 + 1$, we get:

$$\int \frac{x}{x^2+1} \, dx = \frac{1}{2} \ln |x^2 + 1| + C$$

Method B: Partial Fractions

Another method for solving integrals is partial fractions. This involves breaking down a rational function into simpler fractions, which can then be integrated separately. In the case of the integral $\int \frac{x}{x^2+1} \, dx$, we can use partial fractions to rewrite the integral in a more manageable form.

Let's start by factoring the denominator, $x^2 + 1$. This is a quadratic expression that cannot be factored further, so we will leave it in its current form. We can now write the integral as:

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

We can now use partial fractions to break down the rational function into simpler fractions. We can write:

$$\frac{x}{x^2+1} = \frac{A}{x+i} + \frac{B}{x-i}$$

where $A$ and $B$ are constants to be determined. We can now multiply both sides of the equation by $(x^2 + 1)$ to get:

$$x = A(x-i) + B(x+i)$$

We can now expand the right-hand side of the equation and equate coefficients:

$$x = Ax - A i + Bx + Bi$$

Equating coefficients of $x$, we get:

$$1 = A + B$$

Equating coefficients of $i$, we get:

$$0 = -A + B$$

We can now solve this system of equations to find the values of $A$ and $B$:

$$A = \frac{1}{2}$$

$$B = \frac{1}{2}$$

We can now substitute these values back into the partial fractions decomposition:

$$\frac{x}{x^2+1} = \frac{\frac{1}{2}}{x+i} + \frac{\frac{1}{2}}{x-i}$$

We can now integrate this expression with respect to $x$:

$$\int \frac{x}{x^2+1} \, dx = \frac{1}{2} \int \frac{1}{x+i} \, dx + \frac{1}{2} \int \frac{1}{x-i} \, dx$$

We can now integrate each of these expressions separately:

$$\int \frac{1}{x+i} \, dx = \ln |x+i| + C$$

$$\int \frac{1}{x-i} \, dx = \ln |x-i| + C$$

We can now substitute these expressions back into the original integral:

$$\int \frac{x}{x^2+1} \, dx = \frac{1}{2} \ln |x+i| + \frac{1}{2} \ln |x-i| + C$$

Method C: Integration by Parts

Another method for solving integrals is integration by parts. This involves integrating one function and differentiating the other function, and then switching the order of integration and differentiation. In the case of the integral $\int \frac{x}{x^2+1} \, dx$, we can use integration by parts to rewrite the integral in a more manageable form.

Let's start by defining two functions, $u = x$ and $dv = \frac{1}{x^2+1} \, dx$. We can now integrate $dv$ with respect to $x$:

$$\int \frac{1}{x^2+1} \, dx = \arctan x + C$$

We can now differentiate $u$ with respect to $x$:

$$\frac{du}{dx} = 1$$

We can now use integration by parts to rewrite the integral:

$$\int \frac{x}{x^2+1} \, dx = u \int dv - \int v \, du$$

Substituting in the values of $u$ and $dv$, we get:

$$\int \frac{x}{x^2+1} \, dx = x \arctan x - \int \arctan x \, dx$$

We can now integrate the expression on the right-hand side:

$$\int \arctan x \, dx = x \arctan x - \frac{1}{2} \ln (x^2+1) + C$$

We can now substitute this expression back into the original integral:

$$\int \frac{x}{x^2+1} \, dx = x \arctan x - \frac{1}{2} \ln (x^2+1) + C$$

Conclusion

In this article, we have explored three different methods for solving the integral $\int \frac{x}{x^2+1} \, dx$. We have seen that substitution, partial fractions, and integration by parts can all be used to solve this integral. Each of these methods has its own strengths and weaknesses, and the choice of method will depend on the specific characteristics of the integral.

In the case of the integral $\int \frac{x}{x^2+1} \, dx$, we have seen that substitution is the most straightforward method. This involves replacing a portion of the integral with a new variable, which can simplify the expression and make it easier to integrate.

We have also seen that partial fractions can be used to break down the rational function into simpler fractions, which can then be integrated separately. This method can be more complicated than substitution, but it can also be more powerful.

Finally, we have seen that integration by parts can be used to rewrite the integral in a more manageable form. This method involves integrating one function and differentiating the other function, and then switching the order of integration and differentiation.

In conclusion, the integral $\int \frac{x}{x^2+1} \, dx$ can be solved using a variety of methods, including substitution, partial fractions, and integration by parts. Each of these methods has its own strengths and weaknesses, and the choice of method will depend on the specific characteristics of the integral.

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Introduction

In our previous article, we explored three different methods for solving the integral $\int \frac{x}{x^2+1} \, dx$. We saw that substitution, partial fractions, and integration by parts can all be used to solve this integral. In this article, we will answer some common questions that students may have when trying to solve this integral.

Q: What is the most straightforward method for solving the integral $\int \frac{x}{x^2+1} \, dx$?

A: The most straightforward method for solving the integral $\int \frac{x}{x^2+1} \, dx$ is substitution. This involves replacing a portion of the integral with a new variable, which can simplify the expression and make it easier to integrate.

Q: Can partial fractions be used to solve the integral $\int \frac{x}{x^2+1} \, dx$?

A: Yes, partial fractions can be used to solve the integral $\int \frac{x}{x^2+1} \, dx$. This involves breaking down the rational function into simpler fractions, which can then be integrated separately.

Q: Is integration by parts a good method for solving the integral $\int \frac{x}{x^2+1} \, dx$?

A: Integration by parts can be a good method for solving the integral $\int \frac{x}{x^2+1} \, dx$, but it can also be more complicated than substitution or partial fractions. This method involves integrating one function and differentiating the other function, and then switching the order of integration and differentiation.

Q: What are some common mistakes to avoid when trying to solve the integral $\int \frac{x}{x^2+1} \, dx$?

A: Some common mistakes to avoid when trying to solve the integral $\int \frac{x}{x^2+1} \, dx$ include:

• Not simplifying the expression before trying to integrate it
• Not using the correct method for the given integral
• Not checking the work for errors
• Not using a calculator or computer to check the answer

Q: How can I check my work when trying to solve the integral $\int \frac{x}{x^2+1} \, dx$?

A: There are several ways to check your work when trying to solve the integral $\int \frac{x}{x^2+1} \, dx$. These include:

• Using a calculator or computer to check the answer
• Checking the work for errors
• Using a different method to solve the integral
• Checking the answer against a known solution

Q: What are some real-world applications of the integral $\int \frac{x}{x^2+1} \, dx$?

A: The integral $\int \frac{x}{x^2+1} \, dx$ has several real-world applications, including:

• Calculating the area under a curve
• Finding the volume of a solid
• Calculating the work done by a force
• Modeling population growth

Conclusion

In this article, we have answered some common questions that students may have when trying to solve the integral $\int \frac{x}{x^2+1} \, dx$. We have seen that substitution, partial fractions, and integration by parts can all be used to solve this integral, and we have discussed some common mistakes to avoid and ways to check your work. We have also seen some real-world applications of the integral $\int \frac{x}{x^2+1} \, dx$.