Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int\left(5 x^{2}+1\right)\left(5 x^{3}+3 x-8\right)^{6} d x $$

Step-by-Step Solution

Answer

\(\frac{1}{21}(5x^3 + 3x - 8)^7 + C\)

Step 1: Identify the Basis for Substitution

The integrand is of the form \((5x^2+1)(5x^3+3x-8)^6\). Recognize that \((5x^3+3x-8)\) can be set as a substitution variable since its derivative can help simplify the integration process.

Step 2: Define the Substitution Variable

Set \(u = 5x^3 + 3x - 8\). The next step involves finding \(du\).

Step 3: Differentiate to Find du

Differentiate \(u = 5x^3 + 3x - 8\):\[ \frac{du}{dx} = 15x^2 + 3 \]Therefore, \( du = (15x^2 + 3) dx \).

Step 4: Simplify the Expression

Notice that the integrand contains a factor \((5x^2 + 1)\). Factor out and compare with \(du = (15x^2 + 3) dx\). We write:\[ 5x^2 + 1 = \frac{1}{3}(15x^2 + 3) \]Thus, the original integral can be rewritten as:\[ \frac{1}{3} \int (15x^2 + 3)(5x^3 + 3x - 8)^6 dx \]

Step 5: Substitute and Simplify Integral

Make the substitution using \(u\) and \(du\):\[ \frac{1}{3} \int (15x^2 + 3) u^6 \cdot dx = \frac{1}{3} \int u^6 \cdot du \]

Step 6: Integrate Using Power Rule

Apply the power rule for integration:\[ \int u^n du = \frac{u^{n+1}}{n+1} + C \]For \(n=6\), integrate to get:\[ \frac{1}{3} \cdot \frac{u^7}{7} + C = \frac{1}{21} u^7 + C \]

Step 7: Back-Substitute in Terms of x

Replace \(u\) back with \(5x^3 + 3x - 8\):\[ \frac{1}{21}(5x^3 + 3x - 8)^7 + C \]

Step 8: Write the Final Answer

The definite integral is simplified to:\[ \frac{1}{21}(5x^3 + 3x - 8)^7 + C \]

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus used to find the area under a curve over a specific interval. When you are evaluating a definite integral, you have two limits: a lower limit and an upper limit. These limits dictate the range across which you are summing up the infinitesimally small areas between the curve and the x-axis.

The definite integral of a function from a to b is denoted by:\[\int_{a}^{b} f(x) \, dx\]Here,

• \(f(x)\) is the function you are integrating.
• \(a\) and \(b\) are the limits of integration.

The result could be a single numerical value representing the net area.

While working with definite integrals, the Fundamental Theorem of Calculus connects the concept of differentiation with integration, allowing us to evaluate definite integrals by finding an antiderivative. But, remember, unlike indefinite integrals, definite integrals do not include an integration constant, as they result in a number rather than a family of functions.

Power Rule for Integration

The power rule for integration is a straightforward technique used to find the antiderivative or integral of functions with polynomial terms. The formula states:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]which is applicable for any real number \(n\) except \(n = -1\). This formula is quite handy, especially when dealing with polynomials, because it allows us to quickly find integrals.

Let's break down the power rule:

• The variable \(x\) is raised to a power of \(n\).
• You add 1 to the exponent \(n\) to get the new exponent.
• Divide by the new exponent to calculate the coefficient of the term in the integral's expression.

Don't forget the constant \(C\) at the end, particularly when dealing with indefinite integrals. This constant represents the family of all antiderivatives of the function. However, if you are evaluating a definite integral, as mentioned, you focus on the computation of a numerical value and this constant is omitted.

Substitution Variable

In integration, choosing a substitution variable appropriately simplifies the problem significantly through the method known as integration by substitution.

This approach involves replacing a complicated part of the integrand with a single variable, often denoted as \(u\), which helps in transforming the integral into a readily manageable form. The key to successful substitution is ensuring that the derivative of the chosen substitution variable, \(u\), should also be present in the integrand.

Steps in Selecting a Substitution Variable:

• Select a part of the integrand that, when differentiated, also appears within the integrand.
• Define \(u\) based on this selected part, and find \(du\) by differentiating \(u\) with respect to \(x\).
• Rewrite the integral in terms of \(u\) and \(du\), effectively transforming the integral into a simpler form.

Upon solving the integral in terms of \(u\), the last step is to substitute back the original components to express the integral in terms of \(x\). This substitution technique is practical for integrals that align well with the derivatives of their parts, as shown in the original solution.