Problem 4 - Calculus for Biology and Medicine

Question

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 4 x^{3}\left(4-x^{4}\right)^{1 / 3} d x, \text { with } u=4-x^{4} $$

Step-by-Step Solution

Answer

The integral is \(-\frac{3}{4} (4-x^4)^{4/3} + C\).

Step 1: Identify the Substitution

We are given the substitution \( u = 4 - x^4 \). Therefore, we will need to express the integral in terms of \( u \).

Step 2: Differentiate to Find du

Differentiate \( u = 4 - x^4 \) with respect to \( x \):\[ \frac{du}{dx} = -4x^3 \] This gives us \( du = -4x^3 dx \).

Step 3: Solve for dx in Terms of du

Rearrange the differential equation to solve for \( dx \):\[ dx = \frac{du}{-4x^3} \]

Step 4: Substitute in the Integral

Substitute \( u = 4 - x^4 \) and \( dx = \frac{du}{-4x^3} \) into the integral:\[ \int 4x^3(4-x^4)^{1/3} dx = \int 4x^3u^{1/3} \frac{du}{-4x^3} \]

Step 5: Simplify the Integral

Cancel \( 4x^3 \) in the numerator and the denominator:\[ = \int -u^{1/3} \, du \]

Step 6: Integrate with Respect to u

Use the power rule for integration to evaluate the integral:\[ \int -u^{1/3} \, du = -\frac{3}{4} u^{4/3} + C \] where \( C \) is the integration constant.

Step 7: Substitute Back in Terms of x

Substitute back \( u = 4 - x^4 \) into the integrated expression:\[ -\frac{3}{4} (4-x^4)^{4/3} + C \]

Key Concepts

Substitution Method

The substitution method is a versatile and powerful technique in calculus. It helps in solving complex integrals by simplifying them into more manageable forms. The strategy is to make a substitution for part of the integral, turning it into a simpler integral. In the given exercise, we begin with the substitution of a new variable, \( u = 4 - x^4 \). This redefines the integral in terms of \( u \) instead of \( x \), making the integration process easier.
Through this substitution:

• The complicated expression \( (4-x^4)^{1/3} \) becomes \( u^{1/3} \).
• The derivative \( du = -4x^3 dx \) allows us to rewrite \( dx \) in terms of \( du \).

By expressing \( dx \) as \( dx = \frac{du}{-4x^3} \), we can substitute back into the integral, cancel out terms, and focus on a simplified integral of \( u \). This method is crucial for integrals involving compositions of functions, transforming tricky problems into simpler ones by changing variables.

Integration Techniques

Integration techniques are essential tools in calculus, allowing us to find antiderivatives or evaluate integrals that are not immediately obvious. Among these techniques, substitution plays a pivotal role. However, other methods like integration by parts, partial fractions, and trigonometric integrals help tackle a variety of integral problems.
In the exercise, substitution is used because:

• It simplifies the recursion of \( x \) and \( u \) making the integral more feasible.
• Error-prone lengthy computations are avoided as the expression becomes more straightforward.

All complex integration techniques aim to break down integrations into simpler parts that can be easily managed and computed. The understanding of when and how to apply these techniques is fundamental to effectively navigate through various calculus problems, ensuring accurate and efficient solutions.

Power Rule in Integration

The power rule in integration is one of the most straightforward rules and it allows us to integrate polynomials easily. If you have a term of the form \( x^n \), the power rule states that the integral is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
In the provided solution, we transformed the integral into one that involves \( u^{1/3} \). Applying the power rule here involves:

• First, rewriting \( u^{1/3} \) in the form \( u^n \), where \( n = 1/3 \).
• Integrating \( -u^{1/3} \) using the power rule, leading to \( -\frac{3}{4} u^{4/3} + C \).
• The index \( n = 1/3 \) is incremented by 1, giving us \( u^{4/3} \).

This rule is critical for handling polynomial terms or expressions which can be easily expressed as a power of a variable. Mastering the power rule simplifies many integral calculations, making it one of the most foundational techniques in integral calculus.