Problem 3 - Active Calculus

Question

Find the following integral. Note that you can check your answer by differentiation. \(\int \frac{\ln ^{7}(z)}{z} d z=\) _______________________

Step-by-Step Solution

Answer

\( \frac{(\text{ln} (z))^{8}}{8} + C \)

Step 1: Recognize the structure of the integrand

Observe that the integrand \(\frac{\text{ln}^{7}(z)}{z}\) suggests a substitution of the form \(u = \text{ln}(z)\). This will simplify the integral.

Step 2: Make the substitution

Set \(u = \text{ln}(z)\). Then, the differential \(du = \frac{1}{z} dz\) follows directly from this substitution.

Step 3: Substitute and simplify

Replace \( \text{ln}(z) \) with \( u \) and \( \frac{1}{z} dz \) with \( du \) to transform the given integral: \[ \int \frac{\text{ln}^{7}(z)}{z} dz = \int u^{7} du \]

Step 4: Integrate with respect to the new variable

Integrate \(u^{7}\) with respect to \(u\): \[ \int u^{7} du = \frac{u^{8}}{8} + C \]

Step 5: Substitute back the original variable

Re-substitute \( u = \text{ln}(z) \) back into the expression: \[ \frac{(\text{ln}(z))^{8}}{8} + C \]

Key Concepts

logarithmic integration

Logarithmic integration is a technique used when the integrand includes natural logarithms, often in forms like \(\frac{\text{ln}(x)}{x}\). The presence of \(\text{ln}(z)\) in the original integral \(\frac{\text{ln}^{7}(z)}{z} dz\) indicates that this technique will be useful. In our exercise, we used the natural logarithm function, \(\text{ln}(z)\), to set up our substitution, leading us to a much simpler form.
Generally, when you see \(\text{ln}(x)\) in the integrand, it's a hint to use logarithmic integration or substitution methods. This technique helps by turning complicated expressions into simpler polynomials or other forms that are easier to deal with.

substitution method

The substitution method is a powerful tool in calculus for simplifying integrals by replacing a complicated expression with a simpler variable. In our problem, we observed that \(\frac{\text{ln}^{7}(z)}{z} dz\) looked complicated. By setting \(\text{ln}(z) = u\), we converted a difficult integral into an easier one.
Here's how it works:

• Choose a substitution \(u = g(x)\) that simplifies the integrand.
• Find the differential \(du\) corresponding to \(u = g(x)\).
• Rewrite the integral in terms of \(u\).
• Solve the simpler integral.
• Substitute the original variable back in.

In our case, we substituted \(u = \text{ln}(z)\) and \(du = \frac{1}{z} dz\). This converted our integral into \( \int u^{7} du\) which is straightforward to solve. Remember, the substitution method is useful for both definite and indefinite integrals, and is a key technique for simplifying complex calculus problems.

calculus

Calculus is a branch of mathematics that studies continuous change, encompassing both derivatives and integrals. It's foundational for understanding how things change over time and space. Integrals, like the one we solved, are a fundamental concept in calculus.
Integral calculus is focused on the concept of accumulation, such as areas under a curve. In this exercise, we dealt with an indefinite integral, which finds a function whose derivative matches the integrand.
Steps in integration problems often include:

• Identifying a suitable integration technique.
• Applying that technique systematically.
• Simplifying and solving the integral.

In our example, recognizing the structure of \( \frac{\text{ln}^{7}(z)}{z} dz\) suggested substitution. The goal in calculus problems is to turn complexity into simplicity, which we achieved here through a methodical approach. Mastery of calculus techniques opens doors to understanding advanced topics in mathematics, physics, and engineering.