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Magazine Chapter 6: Problem 38
Find the integrals .Check your answers by differentiation. $$\int \frac{(\ln z)^{2}}{z} d z$$
Short Answer
Expert verified
\( \int \frac{(\ln z)^2}{z} \, dz = \frac{(\ln z)^3}{3} + C \).
Step by step solution
01
Identify the Method for Integration
We have the integral \( \int \frac{(\ln z)^2}{z} \, dz \). Notice that this function seems like a candidate for a substitution method, specifically using substitution for \( \ln z \), since its derivative is \( \frac{1}{z} \).
02
Choose the Substitution
Let \( u = \ln z \), which implies \( du = \frac{1}{z} \, dz \). Substituting into the integral, we get \( \int u^2 \, du \).
03
Integrate with Respect to the New Variable
The integral \( \int u^2 \, du \) is a simple power integration. Using the power rule, we integrate to get \( \frac{u^3}{3} + C \), where \( C \) is the constant of integration.
04
Substitute Back the Original Variable
Replace \( u \) with \( \ln z \) to express the integral in terms of \( z \). Thus, we have \( \frac{(\ln z)^3}{3} + C \) as the solution to the original integral.
05
Verify by Differentiation
Differentiate \( \frac{(\ln z)^3}{3} + C \) with respect to \( z \). Applying the chain rule, the derivative is \( \frac{3(\ln z)^2}{3} \cdot \frac{1}{z} = \frac{(\ln z)^2}{z} \), which matches the original integrand, thus verifying the solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Method of Integration
The extit{Method of Integration}, particularly integration by substitution, is an essential concept in calculus that simplifies the process of finding integrals. Imagine you're facing a complex integral that seems difficult to tackle directly. By transforming it into a simpler form, you make it easier to evaluate.
- **Identifying Candidates:** Not every integral needs substitution. Look for functions that contain a composite function, where part of the function is the derivative of another part. This suggests using substitution.
- **Choosing the Right Substitution:** Once you've spotted a potential candidate, choose a substitution that simplifies the integral. For example, in our case, setting \( u = \ln z \) simplifies the integral drastically, because \( du = \frac{1}{z} dz \).
- **Transform and Integrate:** After choosing a substitution, rewrite the integral in terms of the new variable and integrate. This often turns a complicated integral into a straightforward one.
- **Don't Forget to Reverse:** Once integrated, substitute back to express the answer in terms of the original variable.
Differentiation
Differentiation, the process of finding the derivative, is closely linked to integration. Remember, differentiation and integration are inverse processes.
- **The Role of Differentiation:** In the integration context, differentiation helps us verify our results. After finding an integral, differentiating the result should yield the original integrand if done correctly.
- **Easy Verification:** Take our solution, \( \int \frac{(\ln z)^2}{z} dz = \frac{(\ln z)^3}{3} + C \). To verify, differentiate it with respect to \( z \). Applying the chain rule, the derivative simplifies back to our original integrand, which confirms the correctness.
- **Chain Rule in Action:** Here, the chain rule is crucial, as it accounts for derivatives of composite functions. When differentiating \( \frac{(\ln z)^3}{3} \), you first apply the power rule, then multiply by the derivative of \( \ln z \). This process reflects the interconnected nature of differentiation and integration.
Power Rule
The extit{Power Rule} is a fundamental technique in calculus that simplifies both differentiation and integration.
- **For Integration:** When integrating \( x^n \), the power rule assists by stating that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \). It's direct and widely applicable for polynomials and functions that can be expressed in polynomial form.
- **Applying During Substitution:** In our problem, after substituting \( u = \ln z \), we integrated \( u^2 \, du \) using the power rule. This gave us \( \frac{u^3}{3} + C \), streamlining the process significantly.
- **Power Rule for Differentiation:** This rule also applies when differentiating, where \( \frac{d}{dx} x^n = nx^{n-1} \). Understanding this parallel helps in both analytic and numerical calculations.
- **A Simple Tool for Complex Problems:** Despite its simplicity, the power rule is instrumental in solving complex integrals after simplifying them via substitution. It underlies many steps in calculus and acts as a crucial baseline for students mastering integration and differentiation.
