Problem 63 - Calculus: Early Transcendentals

Question

Evaluate the definite integral. \( \displaystyle \int^{13}_0 \frac{dx}{\sqrt[3]{(1 + 2x)^2}} \)

Step-by-Step Solution

Answer

The definite integral evaluates to 3.

Step 1: Identify the Integral Type

The integral to evaluate is \( \int^{13}_0 \frac{dx}{\sqrt[3]{(1 + 2x)^2}} \). This is a definite integral involving a function in the form of a cube root which suggests substitution may simplify the integral.

Step 2: Substitute to Simplify

Let \( u = 1 + 2x \). Then \( du = 2\,dx \) or \( dx = \frac{1}{2}du \). Change the limits of integration: when \( x = 0, u = 1 \) and when \( x = 13, u = 27 \). The integral becomes \( \int^{27}_{1} \frac{1}{2} \cdot \frac{du}{\sqrt[3]{u^2}} \) or \( \frac{1}{2} \int^{27}_{1} u^{-\frac{2}{3}} \, du \).

Step 3: Integrate using Power Rule

The power rule for integration is \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Here, \( n = -\frac{2}{3} \), so the integral of \( u^{-\frac{2}{3}} \) is \( \frac{u^{1/3}}{1/3} \) or \( 3u^{1/3} \). Multiply this result by \( \frac{1}{2} \) from Step 2: \( \frac{3}{2} u^{1/3} \).

Step 4: Evaluate the Definite Integral

Now substitute the limits back into the evaluated antiderivative: \( \left[ \frac{3}{2} u^{1/3} \right]^{27}_{1} \). Calculate each: \( \frac{3}{2} \times 27^{1/3} = \frac{3}{2} \times 3 = \frac{9}{2} \) for the upper limit, and \( \frac{3}{2} \times 1^{1/3} = \frac{3}{2} \) for the lower limit. The result is \( \frac{9}{2} - \frac{3}{2} = 3 \).

Key Concepts

Substitution Method

The substitution method is a powerful tool in calculus for simplifying the integration process, especially when dealing with complex functions. It works by transforming a complicated integral into a simpler one using a change of variable.
The idea is to substitute part of the original function with a new variable to make integration more straightforward.

In this exercise, we encounter the integral \( \int^{13}_0 \frac{dx}{\sqrt[3]{(1 + 2x)^2}} \). The presence of \((1 + 2x)\) suggests using substitution to simplify the function. By letting \( u = 1 + 2x \), we can rewrite the derivative of \( u \) in terms of \( dx \) as \( du = 2\,dx \) or equivalently \( dx = \frac{1}{2}du \). This substitution changes the original integral to a more manageable form:

• When \( x = 0, u = 1 \)
• When \( x = 13, u = 27 \)

This alteration of variables not only simplifies the integrand but also the limits of integration, making it easier to solve. Ultimately, substitution helps you reduce complex integrals into basic forms that are easier to compute.

Integration by Power Rule

The integration by power rule is a key technique for solving integrals of the form \( \int u^n \, du \). This rule states that the integral of \( u^n \) is given by \( \frac{u^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
This formula works well for any real number \( n \) except \( n = -1 \).

In the exercise, after using substitution, we have the integral \( \frac{1}{2} \int^{27}_{1} u^{-\frac{2}{3}} \, du \). Here the exponent \( n \) is \( -\frac{2}{3} \). Using the power rule:

• Calculate the new exponent: add 1 to \( -\frac{2}{3} \) resulting in \( \frac{1}{3} \).
• The corresponding antiderivative becomes \( \frac{u^{1/3}}{1/3} \) which simplifies to \( 3u^{1/3} \).

The multiplication by \( \frac{1}{2} \) from the substitution step gives us \( \frac{3}{2} u^{1/3} \).
The execution of the power rule lets us transform the integrand from a fraction involving roots into a clear algebraic expression, facilitating straightforward calculation of the definite integral.

Cube Root Function

In mathematics, understanding functions involving roots is essential, and in this instance, we consider the cube root function. The cube root function is expressed as \( \sqrt[3]{x} \) and represents a number that when multiplied by itself three times yields the original number \( x \).
Cubing produces a broader range of outputs than square roots, as cube roots include both positive and negative ranges.

In the problem's context, the function \( \sqrt[3]{(1 + 2x)^2} \) shows a nested root that we aim to simplify through substitution. By changing variables to \( u = 1 + 2x \), we simplify this expression to \( u^{-\frac{2}{3}} \). Here the cube root function plays a crucial role in the integration process:

• It impacts how the integral is transformed and solved.
• Cube roots can appear in many growth contexts, emphasizing the importance of simplification in integral solving.

Comprehending the properties of cube roots aids in recognizing possible substitutions and efficient strategies for solving complex integrals.