Problem 61 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \frac{(1-\ln t)^{2}}{t} d t=\) (A) \(\frac{1}{3}(1-\ln t)^{3}+C\) (B) \(\ln t-2 \ln ^{2} t+\ln ^{3} t+C\) (C) \(\ln t-\ln ^{2} t+\frac{\ln t^{3}}{3}+C\) (D) \(-\frac{(1-\ln t)^{3}}{3}+C\)

Step-by-Step Solution

Answer

The solution is (D) \(-\frac{(1-\ln t)^3}{3}+C\).

Step 1: Identify the Substitution

Notice that the integral involves a function of the form \((1 - \ln t)^2 / t\). This suggests using a substitution. Let \( u = 1 - \ln t \). Therefore, \( du = -\frac{1}{t} dt \) or \(-du = \frac{1}{t} dt \).

Step 2: Substitute into the Integral

Substitute \( u = 1 - \ln t \) and \( du = -\frac{1}{t} dt \) into the integral:\[\int \frac{(1 - \ln t)^2}{t} dt = \int u^2 (-du) = -\int u^2 \, du\]

Step 3: Integrate with Respect to u

Perform the integration with respect to \(u\):\[-\int u^2 \, du = -\left(\frac{u^3}{3}\right) + C\]Simplify this to get \(-\frac{u^3}{3} + C\).

Step 4: Substitute Back in Terms of t

Replace \(u\) with \(1 - \ln t\) to transform the indefinite integral back in terms of \(t\):\[-\frac{(1 - \ln t)^3}{3} + C\]This is equivalent to option (D).

Key Concepts

Integration Techniques

Integration techniques are various methods used to find integrals, which are the opposite of derivatives. Understanding these techniques allows us to solve complex calculus problems more efficiently. In the exercise above, we specifically focus on the technique of substitution.

There are several common integration techniques:

• Substitution Method: This is useful when the integral involves a composite function. It simplifies the integral into a more standard form.
• Integration by Parts: This is applied by solving the integral of a product of two functions.
• Partial Fraction Decomposition: Best for rational functions, where the numerator and the denominator are polynomials.
• Trigonometric Integration: Involves integrals with trigonometric functions.

Each technique has specific uses, depending on the form and complexity of the function being integrated.

Substitution Method

The substitution method, or "u-substitution," is an integration technique used to simplify an integral by making a substitution. For the given problem, this method plays a crucial role. It simplifies the problem by transforming it into a more manageable form.

Step-by-step usage of the substitution method involves:

• Identifying a substitution: Look for a function within the integral that, if simplified, could make the integral easier.
• Make the substitution: Replace the identified function with a new variable, often denoted as 'u'. This will change the integral into terms of 'u'.
• Adjust the differential: Rearrange terms to express the differential in terms of the new variable 'u' (e.g., if you let \( u = 1 - \ln t \), then find \( du = -\frac{1}{t} \, dt \)).
• Integrate: Solve the new integral in terms of 'u'.
• Substitute back: Once the integration is complete, replace 'u' back with the original variables.

This approach reduces the complexity of the integral and can often turn an otherwise challenging problem into something straightforward. In the original exercise, we used: \( u = 1 - \ln t \), making it possible to integrate through \( -\int u^2 \, du \).

Indefinite Integrals

Indefinite integrals represent a general form of integration without specific bounds. They are often expressed with a constant of integration, noted as \( C \), indicating the family of functions that differ by a constant.

Here are some key aspects of indefinite integrals:

• General Solution: For a function \( f(x) \), the indefinite integral is represented as \( \int f(x) \, dx \), including \( C \) (i.e., \( F(x) + C \)).
• Finding Antiderivatives: Indefinite integrals involve finding an antiderivative of the function being integrated, which is a fundamental aspect of calculus.
• Integral Form: In this method, the derivative of the antiderivative should equal the original function.

In the example given, the integration of \( \frac{(1 - \ln t)^2}{t} \, dt \) returns an indefinite result \( -\frac{(1-\ln t)^3}{3} + C \), reflecting the formula's versatility covering a range of functions with \( C \) accounting for any constant differences.