Problem 35 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the indefinite integrals. $$ \int x^{3} \sqrt{5+x^{2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{1}{5} (5+x^2)^{5/2} - \frac{5}{3} (5+x^2)^{3/2} + C \).

Step 1: Choose a Substitution

To use substitution, we start by observing the integral and identifying part of it that can be substituted to simplify the expression. Here, a suitable substitution is to let \[ u = 5 + x^2 \]because the derivative of this expression, \ 2x \, dx, \ closely resembles the remaining expression \ x \, dx.\. This substitution will simplify the radical component.

Step 2: Differentiate the Substitution

Next, we need to find the differential \ du.\ Differentiate both sides with respect to \ x.\\[ du = 2x \, dx \]This result will help us replace \ x \, dx \ in the original integral.

Step 3: Solve for x dx in terms of du

From the previous step, solve for \ x \, dx \ in terms of \ du\ to substitute into the integral:\[ x \, dx = \frac{1}{2} du \]

Step 4: Rewrite the Integral in Terms of u

Substitute \ u \ and \ du \ into the original integral:\[ \int x^3 \sqrt{5 + x^2} \, dx = \int x^2 \cdot x\sqrt{u} \, dx = \int x^2 \cdot\frac{1}{2}\sqrt{u} \, du \]Since \ u = 5 + x^2 \we can express \ x^2 = u - 5.\ So the integral becomes:\[ \frac{1}{2} \int (u - 5) \sqrt{u} \, du \]

Step 5: Simplify and Integrate

Distribute and break down the integrand:\[ \frac{1}{2} \int (u^{\frac{3}{2}} - 5u^{\frac{1}{2}}) \, du \]Integrate term by term:\[ \frac{1}{2} \left( \frac{2}{5} u^{\frac{5}{2}} - \frac{10}{3} u^{\frac{3}{2}} \right) + C \]which simplifies to:\[ \frac{1}{5} u^{\frac{5}{2}} - \frac{5}{3} u^{\frac{3}{2}} + C \]

Step 6: Substitute Back in Terms of x

Finally, substitute \ u = 5 + x^2 \ back into the antiderivative to express it in terms of \ x:\\[ \frac{1}{5} (5 + x^2)^{\frac{5}{2}} - \frac{5}{3} (5 + x^2)^{\frac{3}{2}} + C \]

Key Concepts

Indefinite Integrals

In calculus, an indefinite integral, also known as an antiderivative, is a function that reverses differentiation. It aims to find a function whose derivative is the given function.
An indefinite integral is represented by the integral sign followed by a function and a differential, like \( \int f(x) \, dx \). The result, or antiderivative, will include a constant of integration \( C \), to account for the fact that the derivative of a constant is zero.
This constant represents an infinite number of possible antiderivatives since adding or subtracting a constant won’t change the derivative. Therefore, indefinite integrals yield a family of functions. When working with indefinite integrals, keep in mind:

• They are always accompanied by the constant of integration \( C \).
• The purpose is to find the original function from its derivative.
• In problems involving substitution, like the one in this exercise, correctly choosing a substitution can simplify the integration process.

U-Substitution

U-Substitution is a method in calculus that simplifies finding the antiderivatives of composite functions. It works by substituting a part of the integrand (the function being integrated) with a single variable \( u \).
This helps in transforming the integral into a simpler form that's easier to evaluate.This method is particularly effective for integrals involving compositions of functions, like those with nested expressions. To use u-substitution:

• Identify a part of the integrand that could be expressed with a single variable (\( u \)). Often, looking at a composite function or inside of a parenthesis is a good start.
• Calculate the differential \( du \) corresponding to \( u \), enabling the rewriting of the integral in terms of \( du \).
• Replace the chosen part and its differential in the expression to transform the integral into terms of \( u \).
• Evaluate the integral in terms of \( u \) and substitute back to express the antiderivative in terms of the original variable, \( x \).

In this exercise, setting \( u = 5 + x^2 \) simplified the original complex integral into an easier form, making it straightforward to integrate.

Integral Calculus

Integral calculus is a branch of calculus that deals with integration, helping to calculate areas, volumes, and other quantities related to accumulations. It complements differential calculus, which focuses on rates of change.
Understanding integral calculus involves grasping two main types of integrals:

• Indefinite Integrals: These do not have specific limits and represent a family of functions. They look for a function whose derivative will yield the given integrand, ending with a constant of integration \( C \).
• Definite Integrals: These have set upper and lower limits and calculate the total accumulation over an interval, providing a specific numerical result.

Integral calculus is essential for solving problems that involve finding areas under curves, accumulating quantities, and even solving differential equations.
Methods like \( u \)-substitution, as shown in the exercise, are integral part of simplifying complex integrals – helping transform them into more manageable computations and deepening the understanding of integral calculus's practical applications.