Problem 46 - Calculus and its applications

Question

Evaluate. $$ \int_{1}^{2} x\left(x^{2}-1\right)^{7} d x $$

Step-by-Step Solution

Answer

The evaluated integral is 410.0625.

Step 1: Identify the type of integral

The given integral is \( \int_{1}^{2} x(x^2-1)^7 \, dx \). This looks like an integral that can be solved using substitution, because of the form \((x^2-1)^7\). The presence of \(x\) as another factor suggests a potential substitution for \(x^2-1\).

Step 2: Choose the substitution

Let \(u = x^2 - 1\). This implies \(du = 2x \, dx\) or \( x \, dx = \frac{1}{2} \, du \). We'll use this substitution to simplify the integral.

Step 3: Change limits of integration

As \(u = x^2 - 1\), we need to find the new limits. When \(x = 1\), \(u = 1^2 - 1 = 0\). When \(x = 2\), \(u = 2^2 - 1 = 3\). So the new limits in terms of \(u\) are from 0 to 3.

Step 4: Substitute and rewrite the integral

Replace \(x(x^2-1)^7\) with \((x)(u)^7\) and substitute \(x \, dx\) with \(\frac{1}{2} \, du\). The integral becomes: \[ \int_{0}^{3} u^7 \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{0}^{3} u^7 \, du. \]

Step 5: Evaluate the integral

To evaluate \(\frac{1}{2} \int_{0}^{3} u^7 \, du\), first integrate \(u^7\) to get \(\frac{u^8}{8}\). The integral becomes: \[ \frac{1}{2} \left[ \frac{u^8}{8} \right]_{0}^{3}. \]

Step 6: Substitute the limits and simplify

Evaluate \(\frac{1}{2} \cdot \frac{u^8}{8}\) from 0 to 3: \[ \frac{1}{2} \left( \frac{3^8}{8} - \frac{0^8}{8} \right) = \frac{1}{2} \left( \frac{6561}{8} \right). \] Simplifying gives: \[ \frac{6561}{16} = 410.0625. \]

Step 7: Final Result

The final evaluated integral is \(410.0625\). Therefore, \( \int_{1}^{2} x(x^2-1)^7 \, dx = 410.0625. \)

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus to simplify integration problems. When you see an integral that involves a complicated function, like \( (x^2-1)^7 \), combined with its derivative \( x \, dx \), substitution can be a great option. This method involves replacing a part of the integral with a single variable, like \( u \), which simplifies the integration process.
Here's how it works:

• First, identify a part of the integral to substitute. In this case, we choose \( u = x^2 - 1 \).
• Next, differentiate \( u \) to find \( du \). Here, \( du = 2x \, dx \, \) which we'll solve for \( x \, dx \, \) giving \( x \, dx = \frac{1}{2} \, du \).
• Finally, change the integral limits based on the substitution. Replace values: \( x = 1 \rightarrow u = 0 \) and \( x = 2 \rightarrow u = 3 \).

Substitution transforms the integral into a simpler form, allowing more straightforward calculation.

Integration Techniques

Integration techniques are strategies that help solve different types of integral problems. Beyond substitution, these can include integration by parts, partial fractions, and trigonometric identities. Each technique is suited to a specific type of problem.

In the example \( \int_{1}^{2} x(x^2-1)^7 \, dx \), the substitution method reduces the original problem to integrating \( \frac{1}{2} \int_{0}^{3} u^7 \, du \). The function \( u^7 \) is straightforward, requiring basic power rule integration:

• Apply the power rule: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \).
• Here, it becomes \( \frac{u^8}{8} \).

This simply means you add 1 to the exponent and divide by the new exponent. Such techniques make complex-looking integrals manageable.

Calculus Problem Solving

Solving calculus problems involves several systematic approaches and logical thinking. The key to success lies in breaking down problems into manageable steps and recognizing which strategy to apply. For instance, when faced with the integral \( \int_{1}^{2} x(x^2-1)^7 \, dx \, \) knowing when to use substitution can make the evaluation much easier.

After substitution simplifies the integral, calculus problem solving does not stop there:

• Evaluate the integral using calculated changes: \( \frac{1}{2} \int_{0}^{3} u^7 \, du = \frac{1}{2} \times \frac{u^{8}}{8} \bigg|_{0}^{3} \).
• Substitute back the limits: \( u = 0 \) and \( u = 3 \) to find the definite value.
• Simplify results to reach the final evaluated number, which in this case yields \( 410.0625 \).

Successful calculus problem solving uses clear steps, appropriate techniques, and patience to reach the correct answer.