Question

\(\int\left[\mathrm{dx} /\left(\mathrm{x}^{4}+\mathrm{x}^{3}\right)\right]=\left(\mathrm{A} / \mathrm{x}^{2}\right)+(\mathrm{B} / \mathrm{x})+\log |\mathrm{x} /(\mathrm{x}+1)|+\mathrm{c}\) (a) \(\mathrm{A}=(1 / 2), \mathrm{B}=1\) (b) \(\mathrm{A}=1, \mathrm{~B}=(1 / 2)\) (c) \(\mathrm{A}=-(1 / 2), \mathrm{B}=1\) (d) \(\mathrm{A}=-1, \mathrm{~B}=-(1 / 2)\)

Short answer

The correct values for A and B in the given integral are A = (1/2) and B = 1. Therefore, the answer is (a).

Step-by-step solution

Identify the given integral and the desired form

The integral we need to evaluate is: \(\int\frac{dx}{x^4 + x^3}\) The solution form is given as: \(A/x^2 + B/x + \log |x / (x + 1)| + c\)

Perform partial fraction decomposition on the function inside the integral

We need to find partial fraction decomposition of the function \(\frac{1}{x^4+x^3}\). Factor out the common term \(x^3\) and rewrite the expression as: \(\frac{1}{x^3(x + 1)}\) Now, the decomposition will be in the form: \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x + 1}\)

Clear the denominator and simplify the equation

To clear the denominator, we will multiply both sides by \(x^3(x+1)\): \(1 = A(x^2)(x+1) + Bx(x)(x+1) + Cx(x+1) + Dx^3\)

Solve for coefficients A, B, C, and D

We now need to solve for the coefficients A, B, C, and D: Setting x = 0, we get C = 1. Now, the equation becomes: \(1=A(x^2)(x+1)+ Bx(x)(x+1)+x(x+1)+Dx^3\) Take derivatives of both sides with respect to x: \(0=2Ax(x+1)+Ax^2 + B(3x^2 + x) + x + 3Dx^2\) Put x = 0 again to find B: \(0 = B\) Now, the equation becomes: \(1=A(x^2)(x+1)+x(x+1)+Dx^3\) Take derivatives of both sides with respect to x again: \(0=2Ax(x+1)+Ax^2+x+3Dx^2\) Put x = 0 one more time: \(0 = A\) Now we are left with: \(1 = x(x+1) + Dx^3\) Setting x = -1, we can solve for D: \(1=0+D(-1)^3 \Rightarrow D=-1\) Now we have A = 0, B = 0, C = 1, and D = -1.

Integrate the decomposed function and compare it to the given solution form

We can now integrate the decomposed function: \(\int \frac{dx}{x^4 + x^3} = \int \left(\frac{1}{x^3} - \frac{1}{x+1}\right) dx\) Using properties of integrals: \(\int \left(\frac{1}{x^3} - \frac{1}{x+1}\right) dx = \int \frac{1}{x^3} dx - \int \frac{1}{x+1} dx\) Now, integrate each term: \(\int \frac{1}{x^3} dx = -\frac{1}{2x^2} + C_1\) \(\int \frac{1}{x+1} dx = \log|x+1| + C_2\) Together, the integral is: \(-\frac{1}{2x^2} + \log|x+1| + C\) Comparing this to the given solution form: \(A/x^2 + B/x + \log |x / (x + 1)| + c\) We can see that the correct option is (a) A = (1/2) , B = 1.

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a powerful algebraic technique used to simplify complex rational expressions. In essence, it breaks down a complicated fraction into a sum of simpler fractions.
These simpler fractions can be integrated more easily. For example, when dealing with the integral \(\int \frac{dx}{x^4 + x^3}\), the goal is to express \(\frac{1}{x^4 + x^3}\) as a sum of fractions with simpler denominators.

• Start by factorizing the denominator. In this case, factor \(x^3\) out to get \(\frac{1}{x^3(x+1)}\).
• Assume a form for the decomposition such as \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x+1}\).
• Clear the denominator by multiplying through by \(x^3(x+1)\), then solve for coefficients \(A, B, C, D\) through substitution and simplification of the resulting polynomial equations.

This helps to isolate the fractions, each with a simpler form that is manageable for direct integration.

Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus, representing the family of all antiderivatives for a function. They are expressed in the form \(\int f(x)\, dx\) and represent the general solution to the problem of finding a function whose derivative is \(f(x)\).

• These integrals include an arbitrary constant \(C\), because taking the derivative of a constant is zero, leading to multiple valid solutions.
• In the problem \(\int \frac{1}{x^3} dx - \int \frac{1}{x+1} dx\), finding the indefinite integral leads to individual components like \(-\frac{1}{2x^2} + \log|x+1| + C\).

While indefinite integrals give us general solutions, specific solutions can be found by applying initial conditions or contexts.

Techniques of Integration

Integration involves a variety of techniques to solve different forms of integrals. Knowing which technique to apply is key to success in solving integral calculus problems. Below are some vital techniques often used:

• Substitution: Useful when an integral contains a function and its derivative.
• Partial Fraction Decomposition: As seen above, it is used for integrating rational functions by breaking them into simpler fractions.
• By Parts: Based on the product rule, useful for integrals of products of functions.

Understanding these techniques allows one to choose the most efficient method for solving an integral. In this exercise, partial fraction decomposition is crucial as it transforms a complicated function into manageable parts, making the integral solvable.