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Chapter 5: Problem 45

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2} \frac{3}{t} d t$$

Short Answer

Expert verified
Question: Evaluate the definite integral of the function \(\frac{3}{t}\) with respect to \(t\) in the interval \([1, 2]\). Answer: The definite integral of the function \(\frac{3}{t}\) with respect to \(t\) in the interval \([1, 2]\) is \(3\ln|2|\).

Step by step solution

01

Find the antiderivative

We need to find the antiderivative of the function \(f(t) = \frac{3}{t}\). Since \(\frac{d}{dt}(\ln|t|) = \frac{1}{t}\), the antiderivative can be found as follows: $$F(t) = 3\int \frac{1}{t} dt = 3\ln|t| + C$$ Since we will use the Fundamental Theorem of Calculus, we do not need the constant of integration \(C\).
02

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that \(\int_{a}^{b} f(t) dt = F(b) - F(a)\). As we found the antiderivative, we can now plug in the interval limits \(a=1\) and \(b=2\) to find the value of the integral. $$\int_{1}^{2} \frac{3}{t} d t = F(2) - F(1) = 3\ln|2| - 3\ln|1|$$ Since \(\ln|1| = 0\), the answer simplifies to: $$\int_{1}^{2} \frac{3}{t} d t = 3\ln|2|$$

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Most popular questions from this chapter

Additional integrals Use a change of variables to evaluate the following integrals. $$\int_{0}^{2} x^{3} \sqrt{16-x^{4}} d x$$

Multiple substitutions Use two or more substitutions to find the following integrals. $$\int \frac{d x}{\sqrt{1+\sqrt{1+x}}}(\text { Hint: Begin with } u=\sqrt{1+x}\text { .) }$$

More than one way Occasionally, two different substitutions do the job. Use both of the given substitutions to evaluate the following integrals. $$\int_{0}^{1} x \sqrt{x+a} d x ; a>0 \quad(u=\sqrt{x+a} \text { and } u=x+a)$$

Consider the function \(f(x)=x^{2}-4 x\) a. Graph \(f\) on the interval \(x \geq 0\) b. For what value of \(b>0\) is \(\int_{0}^{b} f(x) d x=0 ?\) c. In general, for the function \(f(x)=x^{2}-a x,\) where \(a>0,\) for what value of \(b>0\) (as a function of \(a\) ) is \(\int_{0}^{b} f(x) d x=0 ?\)

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\sqrt{3}} \frac{3 d x}{9+x^{2}}$$
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