Problem 32 - Calculus Early Transcendentals

Question

Use a change of variables to find the following indefinite integrals. Check your work by differentiating. $$\int \frac{8 x+6}{2 x^{2}+3 x} d x$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the function $$\frac{8x+6}{2x^2+3x}$$. Answer: $$\int\frac{8x + 6}{2x^2 + 3x}dx = 2\ln|2x^2 + 3x| + C$$

Step 1: Choose the substitution

Let's choose the substitution: $$u = 2x^2 + 3x$$ Now, differentiate u with respect to x: $$du = (4x + 3)dx$$ Notice that the numerator of the integrand is a constant multiple of the derivative of the denominator. That is: $$8x + 6 = 2(4x + 3)$$ Now, let's solve for dx: $$dx = \frac{du}{(4x + 3)}$$

Step 2: Substitute the variables

Replace x in terms of u and dx with the expressions we derived above: $$\int \frac{8x + 6}{2x^2 + 3x}dx = \int \frac{2(4x+3)}{u}\frac{du}{(4x + 3)}$$ The \((4x + 3)\) terms will cancel out, leaving us with: $$\int 2\frac{du}{u}$$

Step 3: Integrate with respect to u

Now, we can integrate the function: $$\int 2\frac{du}{u} = 2\int\frac{du}{u} = 2\ln|u| + C$$ where C is the constant of integration.

Step 4: Substitute back for x

Now, substitute back for x using our initial substitution: $$2\ln|u| + C = 2\ln|2x^2 + 3x| + C$$ So, our indefinite integral is: $$\int\frac{8x + 6}{2x^2 + 3x}dx = 2\ln|2x^2 + 3x| + C$$

Step 5: Check the solution by differentiation

Now, let's check our solution by taking the derivative: $$\frac{d}{dx}(2\ln|2x^2 + 3x| + C) = \frac{d}{dx}(2\ln(2x^2 + 3x))$$ Using the chain rule: $$= 2\frac{1}{2x^2 + 3x}\cdot(4x+3) = \frac{8x+6}{2x^2+3x}$$ This confirms that our solution is correct, as the derivative of our solution matches the given integrand.

Key Concepts

Change of Variables

In calculus, the technique known as "change of variables" is a powerful method used to simplify integrals, especially when the function appears complex. In simple terms, it involves replacing a part of the integrand (the function to be integrated) with a single variable.

This is often done to make the integral easier to solve. For example, in the given problem, we perform a change of variables by substituting:

• Original expression: \(u = 2x^2 + 3x \)
• After differentiation: \( du = (4x + 3)dx \)

This substitution helps us simplify the integral into a basic form that is manageable to solve analytically. Essentially, we're transforming a complex equation into a simpler one by changing how it's expressed.

Substitution Method

The substitution method, a key integration technique, allows complex functions to be integrated by substituting variables. This method not only simplifies the integration process but also leverages the chain rule in reverse.

Let's look at how it works in our example:

• We start by selecting a substitution where \( u = 2x^2 + 3x \).
• Next, we determine its derivative, \( du = (4x + 3)dx \), which aids in eliminating terms and simplifying expressions.
• This insight allows us to substitute \( dx \) with \( \frac{du}{4x + 3} \), which simplifies our equation by canceling terms.

Recognizing such relationships—a derivative of a part of the numerator corresponding to part of the denominator—enables this simplification, making the integral more approachable.

Derivative Check

Ensuring accuracy in indefinite integrals is crucial, and the derivative check is a reliable method to verify solutions. Once an integral is solved, taking its derivative should return the original function, confirming correctness.

In the resolution for this exercise, the solution achieved was:\[ 2\ln|2x^2 + 3x| + C \]Taking the derivative involves applying the chain rule, which breaks down in the following way:

• Multiply the derivative of the logarithm, \( \frac{1}{2x^2 + 3x} \), by the derivative of its argument, \( 4x + 3 \).
• Thus, we obtain the original function \( \frac{8x+6}{2x^2+3x} \).

Achieving the original function through differentiation indicates the integration process and solution were successful. This derivative check is a vital step in confirming the validity of indefinite integrals.

Integration Techniques

Integration techniques are diverse, offering methods to tackle different forms of integrals. Among these, the substitution method is a commonly used technique, particularly when functions mirror derivative forms.

For the example,

• An integrand like \( \frac{8x+6}{2x^2+3x} \) becomes easier with substitution, simplifying into \( \int 2\frac{du}{u} \).
• This essentially transforms it into the integral of the standard form \( \int \frac{du}{u} \equiv \ln|u| \). This form is straightforward to integrate, leveraging natural logarithms.

Using such techniques and selecting the appropriate method ensures that solutions are found effectively, emphasizing how substitutions open pathways to simpler integrations. This encourages an understanding of underlying patterns, empowering you to solve various indefinite integrals with confidence.