/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 Evaluate. \(\int_{0}^{\ln 2} \... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 49

Evaluate. \(\int_{0}^{\ln 2} \frac{e^{x}}{e^{x}+1} d x\).

Short Answer

Expert verified
The integral \(\int_{0}^{\ln 2} \frac{e^{x}}{e^{x}+1} dx\) can be evaluated by making the substitution \(u = e^x + 1\), changing the limits of integration accordingly, and computing the integral as \(\int_2^3 \frac{1}{u} du\). The final answer is \(\ln\frac{3}{2}\).

Step by step solution

01

Substitute

Let's make the substitution \(u = e^x + 1\). Then, differentiate both sides with respect to x: \[du = e^x dx\]
02

Rewrite the integral

Now, rewrite the integral using the substitution: \[\int \frac{e^x}{e^x + 1} dx = \int \frac{1}{u} du\]
03

Calculate the new limits

We also need to change the limits of integration. First, when \(x = 0\): \[u = e^0 + 1 = 1 + 1 = 2\] Next, when \(x = \ln 2\): \[u = e^{\ln 2} + 1 = 2 + 1 = 3\]
04

Compute the integral

Now, we can compute the integral as follows: \[\int_2^3 \frac{1}{u} du\] The anti-derivative of \(\frac{1}{u}\) is \(\ln |u|\). Using the fundamental theorem of calculus: \[\int_2^3 \frac{1}{u} du = \left[\ln |u|\right]_2^3 = \ln |3| - \ln |2| = \ln\frac{3}{2}\]
05

Write the final answer

The final answer is: \[\int_{0}^{\ln 2} \frac{e^{x}}{e^{x}+1} dx = \ln\frac{3}{2}\]

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Key Concepts

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

Substitution Method
The substitution method is a powerful tool in integral calculus for simplifying complex integrals. It involves changing variables to transform an integral into a simpler form. This is akin to the method of 'u-substitution' often used to solve integrals that are not easily manageable. The key idea is to identify a function within the integral that can be replaced with a new variable.
  • First, choose a substitution. In this case, we chose \(u = e^x + 1\).
  • Next, find the derivative of the substitution function with respect to \(x\). Here, \(du = e^x dx\).
  • Substitute \(u\) into the integral to replace \(x\) and its differential.
This method helps simplify the integral, often reducing it to a basic form you can evaluate with standard techniques. After integrating, you substitute back to the original variable if needed.
Definite Integrals
Definite integrals are a cornerstone of calculus, used to calculate the area under a curve over a given interval. They are represented by an integral sign with upper and lower limits.
  • The exercise here involves calculating a definite integral from \(x = 0\) to \(x = \ln 2\).
  • When performing substitution, it's important to adjust these limits to match your new variable. In this example, the limits change to \(u = 2\) through \(u = 3\).
After substitution, you compute the integral within these new limits, reflecting the area covered by the transformed function between the new bounds.
Fundamental Theorem of Calculus
The fundamental theorem of calculus bridges the gap between differentiation and integration, showing that these two operations are essentially inverses of each other. It provides a way to evaluate definite integrals using antiderivatives.
  • After finding the antiderivative of a function, you can apply the theorem by taking the difference of the antiderivative evaluated at the upper and lower limits of the integral.
  • In this context, the antiderivative of \(\frac{1}{u}\) is \(\ln |u|\).
  • Applying the fundamental theorem of calculus, we evaluate this antiderivative from \(u = 2\) to \(u = 3\), resulting in the value \(\ln\frac{3}{2}\).
This process concludes with the final answer, giving us the value of the original definite integral. It's a crucial theorem that simplifies computing the integral of functions when antiderivatives are known.

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Most popular questions from this chapter

Let \(\Omega\) be the region below the graph of \(y=e^{-x^{2}}\) from \(x=0\) to \(x=1\) (a) Find the volume of the solid generated by revaluing \(\Omega\) about the \(y\) -axis. (b) Form the definite integral that gives the volume of the solid generated by revolving \(\Omega\) about the \(x\) -axis using the disk method. (At this point we cannot carry out the integration.)

Let \(g\) be a function everywhere continuous and not identically zero. Show that if \(f^{\prime}(t)=g(t) f(t)\) for all real \(t,\) then either \(f\) is identically zero or \(f\) does not take on the value zero.

The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon- 12, denoted \(^{12} \mathrm{C}\) (a stable isotope), and carban-14, denoted \(^{14} \mathrm{C}\) (a radioactive isotope). The ratio of the amount of \(^{14} \mathrm{C}\) to the amount of \(^{12} \mathrm{C}\) is essentially constant (approximately \(1 / 10,000\) ). When an organism dies, the amount of \(^{12}\)C Present remains unchanged, but the \(^{14} \mathrm{C}\) decays at a rate proportional to the amount present with a half-life of approximately 5700 years. This change in the amount of \(^{14}\) Relative to the amount of \(^{12} \mathrm{C}\) makes it possible to estimate the time at which the organism lived. A fossil found in an archaeological dig was found to contain \(25 \%\) of the original amount of \(^{14} \mathrm{C}\). What is the approximate age of the fossil?

Use a graphing utility to draw the graph of \(f(x)=\frac{1}{1+x^{2}}\) on [0,10]. (a) Calculate \(\int_{0}^{n} f(x) d x\) for \(n=1000,2500,5000,10,000\). (b) What number are these integrals approaching? (c) Determine the value of $$\lim _{t \rightarrow \infty} \int_{0}^{t} \frac{1}{1+x^{2}} d x$$.

Evaluate. $$\int_{0}^{1 / 2} \frac{1}{\sqrt{3-4 x^{2}}} d x$$.
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