Problem 77 - Calculus Early Transcendentals

Question

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{\sqrt[4]{x+2}+1} ; x+2=u^{4}$$

Step-by-Step Solution

Answer

Give a brief explanation of the process used to evaluate the integral $$\int \frac{dx}{\sqrt[4]{x+2} + 1}$$ We began by performing the substitution \(x+2 = u^4\) and then finding the derivative with respect to \(u\), which was \(dx = 4u^3 du\). After substituting and simplifying the integral, we used polynomial long division to rewrite the integrand as a function of \(u+1\). We then evaluated the integral, obtaining the result in terms of \(u\). Finally, we substituted back the original variable \(x\) to obtain the answer as $$\frac{4}{3}(x+2)^{\frac{3}{4}} - 2(x+2)^{\frac{1}{2}} + C$$.

Step 1: Rewrite the integral in terms of \(u\)

First, let's find the derivative of the substitution: $$\frac{dx}{du}=\frac{d(u^4-2)}{du}$$ Now let's expand the derivative: $$\frac{dx}{du}=4u^{3}$$ Then, we can write \(dx\) in terms of \(du\): $$dx=4u^{3}du$$ We can replace \(\sqrt[4]{x+2}\) with \(u\) since \(u = \sqrt[4]{x+2}\). Now substitute \(u\) and \(dx\) into the integral: $$\int \frac{4u^{3} du}{u + 1}$$

Step 2: Simplify and evaluate the integral

Now that we replaced all the terms in the integral, we only need to evaluate it. We can start by simplifying the integral: $$\int \frac{4u^{3} du}{u + 1} = 4\int \frac{u^{3}}{u+1}du$$ Now, perform a polynomial long division or synthetic division, and rewrite \(u^{3}\) as a function of \(u+1\). The result is \(u^{3}=(u+1)(u^{2}-u+1)-1\). Replace this expression into the integral: $$4\int \frac{(u+1)(u^{2}-u+1)-1}{u+1} du$$ Now we can cancel \(u+1\) term from the numerator and denominator and expand the expression: $$4\int u^{2}-u+1-1 du=4\int u^2-u du$$ Now, we can evaluate the integral: $$4\int u^2-u du = 4\left[\frac{1}{3}u^{3}-\frac{1}{2}u^{2}\right] + C = \frac{4}{3}u^{3}-2u^{2}+C $$

Step 3: Substitute \(x\) back into the answer

Now, let's rewrite our solution in terms of \(x\). Recall that we used the substitution \(x+2=u^{4}\). First, solve for \(u\): $$u = \sqrt[4]{x+2}$$ Now substitute this expression for \(u\) into our solution: $$\frac{4}{3}(\sqrt[4]{x+2})^{3} - 2(\sqrt[4]{x+2})^{2} + C$$ This is the final answer: $$\frac{4}{3}(x+2)^{\frac{3}{4}} - 2(x+2)^{\frac{1}{2}} + C$$

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving integrals, especially when dealing with expressions that are hard to integrate directly. This technique involves replacing a complicated part of the integrand with a simpler variable to make the integral easier to solve.
This is particularly useful in cases where the function inside the integral can be transformed into a more familiar form.To apply the substitution method:

• Choose a substitution that simplifies the integrand.
• Express the derivatives to adjust for the substitution.
• Rewrite the entire integral in terms of the new variable.

In the exercise, the substitution \(x+2 = u^{4}\) was chosen because it simplifies the radical expression, allowing us to work with a rational function instead. Once the substitution is made, regressing back to the original variable after integration completes the solution.

Polynomial Long Division

Polynomial long division is a technique used to divide one polynomial by another. This method helps in breaking down complex polynomial expressions into simpler parts, which can then be integrated separately when working with integrals.When applied:

• Divide the terms of the numerator by those of the divisor, starting with the highest degree.
• Multiply, subtract, then repeat with the remainder until all terms are divided.

In the given solution, polynomial long division was used to rewrite \(u^{3}\) as \((u+1)(u^{2}-u+1) - 1\) which simplifies integration. By turning the original complex rational expression into a sum of simpler terms, integration becomes straightforward.

Simplifying Integrals

Simplifying integrals is an essential part of integration to ease the process of finding the antiderivative. It involves algebraic manipulation of the integrand to transform it into a simpler form that is easier to integrate.
This process often employs techniques like polynomial long division, factoring, or other algebraic simplifications.In the problem, the integral \( \int \frac{4u^{3} du}{u + 1} \) was simplified by performing polynomial long division. This step transformed it into the sum \(4\int(u^2-u)du\).
By removing the complex fraction, the task of finding the antiderivative becomes straightforward, leading to a simpler solution.

Rational Function

Rational functions are fractions where both the numerator and the denominator are polynomials. Integration of rational functions often requires methods like substitution and partial fraction decomposition to simplify them into more manageable forms.For instance:

• Express the rational function as a sum or difference of simpler fractions if possible.
• Use algebraic manipulation techniques, such as polynomial long division, to simplify.

In the original problem, the substitution \(x+2=u^{4}\) transformed the integral into one involving a rational function \(\frac{4u^{3}}{u+1}\). The solution relied on simplifying this rational function to make the integration viable, using polynomial long division to reduce it into simpler polynomial parts. This simplification step is crucial for efficient and effective integration of complex rational functions.