Evaluate The Integral:${ \int \frac{10x^2 - 24x + 12}{5x - 2} , Dx }$

Introduction

In this article, we will delve into the world of calculus and explore the process of evaluating a specific integral. The integral in question is ${ \int \frac{10x^2 - 24x + 12}{5x - 2} , dx }$. 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 Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the numerator is $10x^2 - 24x + 12$, and the denominator is $5x - 2$. Rational functions can be evaluated using various techniques, including partial fraction decomposition, substitution, and integration by parts.

Partial Fraction Decomposition

One of the most powerful techniques for evaluating rational functions is partial fraction decomposition. This method involves breaking down the rational function into simpler fractions, which can then be integrated separately. To apply partial fraction decomposition, we need to factor the denominator and express the rational function as a sum of simpler fractions.

Factoring the Denominator

The denominator of the given rational function is $5x - 2$. We can factor this expression as $(5x - 2)$. However, this is not a product of two binomials, so we cannot apply the standard method of factoring. Nevertheless, we can still use partial fraction decomposition to evaluate the integral.

Applying Partial Fraction Decomposition

To apply partial fraction decomposition, we need to express the rational function as a sum of simpler fractions. We can do this by writing:

$$\frac{10x^2 - 24x + 12}{5x - 2} = \frac{A}{5x - 2} + \frac{B}{1}$$

where $A$ and $B$ are constants to be determined.

Finding the Constants

To find the values of $A$ and $B$, we can multiply both sides of the equation by the common denominator, $5x - 2$. This gives us:

$$10x^2 - 24x + 12 = A(1) + B(5x - 2)$$

We can now equate the coefficients of like terms on both sides of the equation. This gives us two equations:

$$10x^2 - 24x + 12 = A + 5Bx - 2B$$

Equating the coefficients of $x^2$, we get:

$$10 = 0$$

This equation is not possible, so we must have made an error in our previous steps. Let's go back and re-examine our work.

Re-examining the Work

Upon re-examining our work, we realize that we made a mistake in factoring the denominator. The correct factorization is $(5x - 2)(x - 1)$. We can now apply partial fraction decomposition as follows:

$$\frac{10x^2 - 24x + 12}{5x - 2} = \frac{A}{5x - 2} + \frac{B}{x - 1}$$

Finding the Constants

To find the values of $A$ and $B$, we can multiply both sides of the equation by the common denominator, $(5x - 2)(x - 1)$. This gives us:

$$10x^2 - 24x + 12 = A(x - 1) + B(5x - 2)$$

We can now equate the coefficients of like terms on both sides of the equation. This gives us two equations:

$$10x^2 - 24x + 12 = Ax - A + 5Bx - 2B$$

Equating the coefficients of $x^2$, we get:

$$10 = A + 5B$$

Equating the coefficients of $x$, we get:

$$-24 = -A + 5B$$

We can now solve these two equations simultaneously to find the values of $A$ and $B$.

Solving for A and B

We can solve the two equations simultaneously by multiplying the first equation by $5$ and the second equation by $-1$. This gives us:

$$50 = 5A + 25B$$

$$24 = A - 5B$$

We can now add these two equations to eliminate the variable $B$. This gives us:

$$74 = 6A$$

We can now solve for $A$ by dividing both sides of the equation by $6$. This gives us:

$$A = \frac{74}{6} = \frac{37}{3}$$

We can now substitute this value of $A$ into one of the original equations to solve for $B$. Let's use the second equation:

$$24 = A - 5B$$

Substituting $A = \frac{37}{3}$, we get:

$$24 = \frac{37}{3} - 5B$$

We can now solve for $B$ by multiplying both sides of the equation by $3$ and then subtracting $\frac{37}{3}$ from both sides. This gives us:

$$72 = 37 - 15B$$

Subtracting $37$ from both sides, we get:

$$35 = -15B$$

Dividing both sides by $-15$, we get:

$$B = -\frac{35}{15} = -\frac{7}{3}$$

Evaluating the Integral

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

$$\int \frac{10x^2 - 24x + 12}{5x - 2} \, dx = \int \left( \frac{\frac{37}{3}}{5x - 2} - \frac{\frac{7}{3}}{x - 1} \right) \, dx$$

We can now integrate each term separately:

$$\int \left( \frac{\frac{37}{3}}{5x - 2} - \frac{\frac{7}{3}}{x - 1} \right) \, dx = \frac{37}{3} \int \frac{1}{5x - 2} \, dx - \frac{7}{3} \int \frac{1}{x - 1} \, dx$$

We can now evaluate each integral separately:

$$\frac{37}{3} \int \frac{1}{5x - 2} \, dx = \frac{37}{3} \ln |5x - 2| + C$$

$$-\frac{7}{3} \int \frac{1}{x - 1} \, dx = -\frac{7}{3} \ln |x - 1| + C$$

We can now combine these two results to get the final answer:

$$\int \frac{10x^2 - 24x + 12}{5x - 2} \, dx = \frac{37}{3} \ln |5x - 2| - \frac{7}{3} \ln |x - 1| + C$$

Conclusion

In this article, we evaluated the integral ${ \int \frac{10x^2 - 24x + 12}{5x - 2} , dx }$. We used partial fraction decomposition to break down the rational function into simpler fractions, which we then integrated separately. The final answer is $\frac{37}{3} \ln |5x - 2| - \frac{7}{3} \ln |x - 1| + C$.

Introduction

In our previous article, we evaluated the integral ${ \int \frac{10x^2 - 24x + 12}{5x - 2} , dx }$. We used partial fraction decomposition to break down the rational function into simpler fractions, which we then integrated separately. In this article, we will answer some common questions related to the evaluation of this integral.

Q: What is partial fraction decomposition?

A: Partial fraction decomposition is a technique used to break down a rational function into simpler fractions. It involves expressing the rational function as a sum of simpler fractions, which can then be integrated separately.

Q: How do I apply partial fraction decomposition?

A: To apply partial fraction decomposition, you need to factor the denominator and express the rational function as a sum of simpler fractions. You can then equate the coefficients of like terms on both sides of the equation to find the values of the constants.

Q: What are the steps involved in partial fraction decomposition?

A: The steps involved in partial fraction decomposition are:

1. Factor the denominator
2. Express the rational function as a sum of simpler fractions
3. Equate the coefficients of like terms on both sides of the equation
4. Solve for the values of the constants

Q: What are some common mistakes to avoid when applying partial fraction decomposition?

A: Some common mistakes to avoid when applying partial fraction decomposition include:

• Not factoring the denominator correctly
• Not expressing the rational function as a sum of simpler fractions correctly
• Not equating the coefficients of like terms on both sides of the equation correctly
• Not solving for the values of the constants correctly

Q: Can I use partial fraction decomposition to evaluate any type of integral?

A: No, partial fraction decomposition can only be used to evaluate integrals of rational functions. It is not applicable to integrals of other types of functions, such as trigonometric functions or exponential functions.

Q: What are some other techniques for evaluating integrals?

A: Some other techniques for evaluating integrals include:

• Substitution
• Integration by parts
• Integration by partial fractions
• Trigonometric substitution
• Integration by reduction formula

Q: How do I choose the right technique for evaluating an integral?

A: To choose the right technique for evaluating an integral, you need to consider the type of function being integrated and the complexity of the integral. You can then choose the technique that is most suitable for the problem at hand.

Q: What are some common applications of integral calculus?

A: Some common applications of integral calculus include:

• Finding the area under curves
• Finding the volume of solids
• Finding the surface area of solids
• Finding the work done by a force
• Finding the center of mass of an object

Q: Why is integral calculus important?

A: Integral calculus is important because it provides a powerful tool for solving problems in a wide range of fields, including physics, engineering, economics, and computer science. It is used to model real-world phenomena and to make predictions about the behavior of complex systems.

Conclusion

In this article, we answered some common questions related to the evaluation of the integral ${ \int \frac{10x^2 - 24x + 12}{5x - 2} , dx }$. We discussed the technique of partial fraction decomposition and provided some tips and tricks for applying it correctly. We also discussed some other techniques for evaluating integrals and provided some common applications of integral calculus.