Problem 22 - Calculus and its applications

Question

Find each integral. $$ \int\left(\frac{4}{x^{3}}+\frac{7}{x}\right) d x $$

Step-by-Step Solution

Answer

The integral is \(-2x^{-2} + 7 \ln |x| + C\).

Step 1: Break Down the Integral

The given integral is \( \int \left( \frac{4}{x^3} + \frac{7}{x} \right) \, dx \). We can break it down into simpler integrals: \( \int \frac{4}{x^3} \, dx + \int \frac{7}{x} \, dx \).

Step 2: Simplify Inside the Integral

Rewrite \( \frac{4}{x^3} \) as \( 4x^{-3} \), so the first integral becomes \( \int 4x^{-3} \, dx \). The second integral \( \frac{7}{x} \) can be rewritten using the property \( \int \frac{1}{x} \, dx = \ln |x| \).

Step 3: Integrate the First Integral using Power Rule

For \( \int 4x^{-3} \, dx \), use the power rule \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), resulting in \( \frac{4}{-2}x^{-2} + C_1 = -2x^{-2} + C_1 \), where \( C_1 \) is a constant of integration.

Step 4: Integrate the Second Integral using Logarithm Rule

For \( \int \frac{7}{x} \, dx \), it directly integrates to \( 7 \ln |x| + C_2 \), because of the rule \( \int \frac{1}{x} \, dx = \ln |x| \).

Step 5: Combine Results

Combine the results of the two integrals: \( -2x^{-2} + 7 \ln |x| + C \), where \( C = C_1 + C_2 \) is the constant of integration.

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding the integral of a function in the form of \( x^n \). It states that when you integrate \( x^n \), you add 1 to the exponent \( n \) and divide by the new exponent: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \( C \) is the constant of integration.
In the exercise, to integrate \( 4x^{-3} \), the exponent \(-3\) is increased by 1 to become \(-2\), and then you divide by this new exponent: \[\int 4x^{-3} \, dx = \frac{4}{-2}x^{-2} = -2x^{-2}\]Don't forget that handling negative exponents involves flipping them into a fraction when expressing the final answer in terms of standard notation.

Logarithm Rule

The logarithm rule is particularly useful when integrating expressions involving \( \frac{1}{x} \). This rule asserts that the integral of \( \frac{1}{x} \) is the natural logarithm of the absolute value of \( x \): \[\int \frac{1}{x} \, dx = \ln |x| + C\]For this exercise, the integral \( \int \frac{7}{x} \, dx \) can directly use this rule. Multiply \( 7 \) by the obtained integral to get: \[\int \frac{7}{x} \, dx = 7 \ln |x| + C_2\]As natural logarithms require positive arguments, hence the absolute value around \( x \). This keeps the expression valid irrespective of the sign of \( x \). The integration constant ensures any possible vertical shifts of the antiderivative function.

Constants of Integration

Constants of integration are added to the result of an indefinite integral to account for any constant term that could have differentiated to zero. When integrating, the derivative of a constant is 0, meaning any number of different antiderivatives could have produced the same function before differentiation.
While solving, multiple integrals could produce separate constants:\( C_1 \) and \( C_2 \). When combining these to find a complete solution, a single constant \( C \) is used: \[C = C_1 + C_2\]This combined constant \( C \) is added to the final expression: \[\int \left( \frac{4}{x^3} + \frac{7}{x} \right) \, dx = -2x^{-2} + 7 \ln |x| + C\]Understanding this concept ensures clarity that every antiderivative is part of a family of functions differing only by a constant.