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Chapter 17: Problem 3

Find $$\int_{a}^{b} \frac{f^{\prime}(t)}{f(t)} d t$$ (for \(f>0 \text { on }[a, b])\).

Short Answer

Expert verified
The integral evaluates to \( \ln\left|\frac{f(b)}{f(a)}\right| \).

Step by step solution

01

Recognize the integral form

Identify that the integral \( \int_{a}^{b} \frac{f^{\prime}(t)}{f(t)} d t \) resembles the natural logarithm derivative form. Notice that \( \frac{d}{dt} [\ln|f(t)|] = \frac{f^{\prime}(t)}{f(t)} \).
02

Apply the substitution

Use the substitution \( u = \ln|f(t)| \). Thus, \( du = \frac{f^{\prime}(t)}{f(t)} dt \). The integral becomes \( \int_{a}^{b} du \).
03

Evaluate the integral

Since \( du = \frac{f^{\prime}(t)}{f(t)} dt \), the integral \( \int_{a}^{b} du = \left. u \right|_{a}^{b} = u(b) - u(a) \). Re-substituting \( u = \ln|f(t)| \), the result is \( \ln|f(b)| - \ln|f(a)| = \ln\left|\frac{f(b)}{f(a)}\right| \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}(x)\), is a fundamental concept in calculus and mathematics. It is the inverse function of the exponential function \(e^x\). In simpler terms, if \(y = \text{ln}(x)\), then \(e^y = x\). The natural logarithm has several important properties:
  • \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \)
  • \( \text{ln}\bigg(\frac{a}{b}\bigg) = \text{ln}(a) - \text{ln}(b) \)
  • \( \text{ln}(a^b) = b \times \text{ln}(a) \)
  • \( \text{ln}(1) = 0 \)
When working with integrals involving logarithms, recognizing these properties can greatly simplify the problem.
Substitution Method
The substitution method, also known as \(\text{substitution rule or change of variables}\), is a technique for evaluating integrals. It simplifies an integral by introducing a new variable. This method is particularly useful when dealing with complex functions. To perform substitution:
  • Select a substitution \(u = g(x)\) where \(g(x)\) is a function of \(x\).
  • Compute the differential \(du = g'(x)dx\).
  • Rewrite the integral in terms of the new variable \(u\).
  • Integrate with respect to \(u\).
  • Finally, substitute back the original variable.
In our given exercise, we identify the function under the integral resembles the derivative of the natural logarithm. By letting \(u = \text{ln}|f(t)|\), we transform the integral into a simpler form.
Definite Integral
A definite integral, represented as \( \int_{a}^{b} f(x) dx \), calculates the net area under a curve \(f(x)\) from \(x = a\) to \(x = b\). Here's a quick summary of its key properties:
  • The definite integral combines both integration and limits of integration.
  • It is evaluated as the difference of the antiderivative of the function at the upper and lower limits.
  • Fundamental Theorem of Calculus connects the definite integral and the antiderivative: \[\int_{a}^{b} f(x) dx = F(b) - F(a) \] where \(F(x)\) is the antiderivative of \(f(x)\).
In the exercise, once we perform the substitution, the integral simplifies. We then apply the limits of integration and utilize the properties of the natural logarithm for the final result.

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Most popular questions from this chapter

Prove that \(\log _{10} 2\) is irrational.

(a) Evaluate \(\lim _{x \rightarrow \infty} e^{-x^{2}} \int_{0}^{x} e^{t^{2}} d t .\) (You should be able to make an educated guess before doing any calculations.) (b) Evaluate the following limits. (i) \(\lim _{x \rightarrow \infty} e^{-x^{2}} \int_{x}^{x+(1 / x)} e^{t^{2}} d t\). (ii) \(\lim _{x \rightarrow \infty} e^{-x^{2}} \int_{x}^{x+\log x) / x} e^{t^{2}} d t\). (iii) \(\lim _{x \rightarrow \infty} e^{-x^{2}} \int_{x}^{x+\log x / 2 x} e^{t^{2}} d t\).

(a) The derivative of \(\log \circ f\) is \(f^{\prime} / f\) This expression is called the logarithmic derivative of \(f .\) It is often easier to compute than \(f^{\prime},\) since products and powers in the expression for \(f\) become sums and products in the expression for log of. The derivative \(f^{\prime}\) can then be recovered simply by multiplying by \(f ;\) this process is called logarithmic differentiation. (b) Use logarithmic differentiation to find \(f^{\prime}(x)\) for each of the following. (i) \(f(x)=(1+x)\left(1+e^{x^{2}}\right)\). (ii) \(f(x)=\frac{(3-x)^{1 / 3} x^{2}}{(1-x)(3+x)^{2 / 3}}\). (iii) \(f(x)=(\sin x)^{\cos x}+(\cos x)^{\sin x}\). (iv) \(f(x)=\frac{e^{x}-e^{-x}}{e^{2 x}\left(1+x^{3}\right)}\).

(a) Let \(f(x)=\log |x|\) for \(x \neq 0 .\) Prove that \(f^{\prime}(x)=1 / x\) for \(x \neq 0\). (b) If \(f(x) \neq 0\) for all \(x,\) prove that \((\log |f|)^{\prime}=f^{\prime} / f\).

Suppose \(f\) satisfies \(f^{\prime}=f\) and \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y .\) Prove that \(f=\exp\) or \(f=0\).
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