Problem 80 - Calculus Single Variable

Question

Integrate by parts repeatedly, each time with \(u\) equal to a power of \(1-x,\) to prove that $$ \int_{0}^{1} \sqrt{x}(1-x)^{4} d x=\frac{8}{3} \cdot \frac{6}{5} \cdot \frac{4}{7} \cdot \frac{2}{9} \cdot \frac{2}{11} $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{8}{3} \cdot \frac{6}{5} \cdot \frac{4}{7} \cdot \frac{2}{9} \cdot \frac{2}{11} \).

Step 1: Set up integration by parts

The integration by parts formula is \( \int u \, dv = uv - \int v \, du \). Let's choose \( u = (1-x)^4 \) and \( dv = \sqrt{x} \, dx \). Calculate \( du \) and \( v \).

Step 2: Find derivatives and integrals

Differentiate \( u \) to get \( du = -4(1-x)^3 \, dx \). Integrate \( dv \) to get \( v = \frac{2}{3} x^{3/2} \). Substitute these into the integration by parts formula.

Step 3: Apply integration by parts

Using integration by parts, calculate \( uv - \int v \, du \). The calculation becomes \( \frac{2}{3} x^{3/2} (1-x)^4 \Big|_0^1 - \int_0^1 \frac{2}{3} x^{3/2} \cdot (-4) (1-x)^3 \, dx \). Evaluate the boundary terms.

Step 4: Simplify the expression further

Evaluate the boundary term, \( \left. \frac{2}{3} x^{3/2} (1-x)^4 \right|_0^1 \), which is 0 because at x=1, \((1-x)^4 = 0\) and at x=0, \(x^{3/2} = 0\). This simplifies our expression to \( \frac{8}{3} \int_0^1 x^{3/2} (1-x)^3 \, dx \).

Step 5: Repeat integration by parts

Repeat the integration by parts on \( \int x^{3/2} (1-x)^3 \, dx \) with \( u = (1-x)^3 \) and \( dv = x^{3/2} \, dx \). Continue this process iteratively.

Step 6: Compute the integral using the pattern

After multiple integrations by parts, observe the pattern formed by the fractions. Notice the terms align with the product \( \frac{8}{3} \cdot \frac{6}{5} \cdot \frac{4}{7} \cdot \frac{2}{9} \cdot \frac{2}{11} \).

Step 7: Conclude with the final result

As you repeatedly apply integration by parts, each new substitution step simplifies the expression, allowing us to confirm the pattern's correctness. Use each resulting integration to establish that: \[ \int_0^1 \sqrt{x}(1-x)^4 \, dx = \frac{8}{3} \cdot \frac{6}{5} \cdot \frac{4}{7} \cdot \frac{2}{9} \cdot \frac{2}{11} \].

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus, allowing us to calculate the area under a curve between two specific points. In this problem, the definite integral \[ \int_{0}^{1} \sqrt{x}(1-x)^{4} \, dx \] spans from 0 to 1 on the x-axis.

Using definite integrals helps us to precisely determine the total area, unlike indefinite integrals, which represent a family of functions. The definite integral's boundaries are crucial, as they limit the evaluation of the area to the specified interval. Key point: When evaluating such integrals, it's essential to understand how the function behaves within these bounds to ensure the correct calculation. This involves carefully selecting and calculating each part of the integrand within the given limits.

Step-by-Step Integration

Step-by-step integration is a systematic method used to solve integrals, especially complex ones like this problem, through a series of calculated steps. Each step simplifies the integral further, working towards the full solution.

For instance, integration by parts, based on the formula \[ \int u \, dv = uv - \int v \, du \] is applied here. In the initial step, we start by identifying parts of the integral that can be expressed as \( u \) and \( dv \). This systematic breakdown is essential:

• Calculate \( du \) from \( u = (1-x)^4 \) by differentiation.
• Determine \( v \) from \( dv = \sqrt{x} \, dx \) by integration.
• Substitute these components back into the formula to simplify the integral.

This procedure is repeated until reaching the desired solution, highlighting how structured and repetitive approaches aid in tackling complex integrals.

Pattern Recognition

Pattern recognition is an insightful strategy in calculus, important for determining solutions more efficiently once a recognizable form emerges. In this integral, a pattern of multiplying fractions develops through repeated application of the integration by parts technique.

Starting with the first partial solution \( \frac{8}{3} \int_0^1 x^{3/2} (1-x)^3 \, dx \), repeated integrations yield a consistent form.
The fractions observed - \( \frac{8}{3} \), \( \frac{6}{5} \), and so on - follow a specific sequence resulting from the various reductions in power of \( (1-x) \).

• This recognition allows for strengthened intuition in predicting further results.
• It highlights the natural structure emerging from iterated mathematical procedures.

With each step, understanding the pattern enhances not just computational efficiency but also deepens comprehension of how integrals tend to behave in similar cases.

Mathematical Proofs

Mathematical proofs provide a way to confirm and validate results of integrals or any mathematical assertions. Here, through repetitive execution of integration by parts, the accuracy of the integral's expression is proven.

By proving that \[ \int_0^1 \sqrt{x}(1-x)^4 \, dx = \frac{8}{3} \cdot \frac{6}{5} \cdot \frac{4}{7} \cdot \frac{2}{9} \cdot \frac{2}{11} \] we ensure correctness and reliability of the solution:

• Each application of integration by parts is justified and reinforces the validity of each step.
• The logical flow from assumptions to conclusions provides clarity and solidification of the result.
• Proofs like these not only cement understanding but also enhance problem-solving skills by verifying every component mathematically.

Such thorough proofs become essential in more advanced mathematical or theoretical explorations, where precision and correctness cannot be compromised.