Problem 44 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int x \sqrt{4-x} d x$$

Step-by-Step Solution

Answer

The integral is \(-\frac{8}{3}(4-x)^{3/2} + \frac{2}{5}(4-x)^{5/2} + C\).

Step 1: Identify the Integration Method

The integral \( \int x \sqrt{4-x} \, dx \) includes a square root function that seems suitable for substitution. Finding a suitable substitution will simplify the integration process.

Step 2: Choose and Apply a Substitution

Let \( u = 4-x \). Then, \( du = -dx \), and \( x = 4-u \). Substitute these into the integral: \[\int x \sqrt{4-x} \, dx = \int (4-u) \sqrt{u} (-du).\] This can be rewritten as \(- \int (4-u) \sqrt{u} \, du\).

Step 3: Simplify the Integral

Distribute \( \sqrt{u} \) in the expression \( - \int (4-u) \sqrt{u} \, du \): \[- \int (4 \sqrt{u} - u \sqrt{u}) \, du = - \int (4u^{1/2} - u^{3/2}) \, du.\] Split into two separate integrals: \[\- \left( \int 4u^{1/2} \, du - \int u^{3/2} \, du \right).\]

Step 4: Integrate Each Term

Integrate each term separately:1. The integral of \( 4u^{1/2} \) is: \( 4 \int u^{1/2} \, du = 4 \cdot \frac{2}{3}u^{3/2} = \frac{8}{3}u^{3/2}\).2. The integral of \( u^{3/2} \) is: \( \int u^{3/2} \, du = \frac{2}{5}u^{5/2} \).Thus, the integrals combine to: \[-\frac{8}{3}u^{3/2} + \frac{2}{5}u^{5/2}.\]

Step 5: Substitute Back in Terms of x

Recall that \( u = 4-x \). Substitute back into the integrated function: \[-\frac{8}{3}(4-x)^{3/2} + \frac{2}{5}(4-x)^{5/2}.\]

Step 6: Write the Final Answer and Include Constant of Integration

The evaluated integral, including the constant of integration \( C \), is:\[\int x \sqrt{4-x} \, dx = -\frac{8}{3}(4-x)^{3/2} + \frac{2}{5}(4-x)^{5/2} + C.\]

Key Concepts

Definite Integrals

A definite integral is a way to calculate the total accumulation of a quantity, often represented as the area under a curve. It is defined between two limits, say from \( a \) to \( b \). When evaluating a definite integral, it's important to:

• Note the limits of integration, which provide the bounds of the interval.
• Use the Fundamental Theorem of Calculus, which links the concept of differentiation with integration.
• Compute the antiderivative of the function within the limits and find the difference of its values at those two points.

Understanding definite integrals requires practice. Applications can range from finding areas to solving complex real-world problems, making them a vital part of calculus.

Integration Techniques

Integration techniques refer to various methods used to solve integrals, which are not always straightforward. Sometimes, you encounter functions where standard methods like finding an antiderivative directly are challenging. Common techniques include:

• Substitution: Used when integral contains a function and its derivative.
• Integration by parts: Useful for integrals of products of functions.
• Partial fraction decomposition: Ideal for rational functions.
• Trigonometric substitution: Effective for trigonometric integrals.

In our problem, substitution is the chosen technique. It's crucial to pick the right method to simplify complex integrals, making them easier to compute.

Variable Substitution

Variable substitution, also known as "u-substitution," is a powerful technique that simplifies complex integrals by transforming them into a more straightforward form. Here’s how it works:

• Choose a substitution \( u = g(x) \), which simplifies the integral.
• Compute \( du \), the derivative of \( u \), in terms of \( dx \).
• Substitute \( u \) and \( du \) back into the integral, converting it entirely in terms of \( u \).
• Carry out the integration in terms of \( u \).
• Finally, substitute back to express your solution in terms of the original variable \( x \).

This technique is especially useful for dealing with integrals that contain functions inside other functions, as seen in our exercise where \( u = 4-x \) simplifies the square root expression, allowing for easier integration.