Problem 51 Evaluate. $$\int_{0}^{\ln 2} \... [FREE SOLUTION]

91影视

Calculus: One and Several Variables

Saturnino L. Salas, Garret J. Etgen, Einar Hille

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 51

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

Short Answer

Expert verified

The short answer, after evaluating the integral, is: $\int_{0}^{\ln 2} \frac{e^{x}}{1+e^{2 x}} d x = \frac{1}{2}\ln\left(\frac{5}{2}\right)$.

Step by step solution

Let's perform the substitution $u = e^x$, so $u(0)=1$, $u(\ln 2)=2$, and $d u= e^x d x$. This will make the denominator look like a more recognizable expression: a quadratic form. #Step 2: Rewrite the integral in terms of u#

Substituting to obtain the integral in terms of $u$: $$\int_{1}^{2} \frac{u}{1+u^2} d u$$ #Step 3: Integration by substitution#

Now we can solve this integral by further substitution. Let $v = u^2$, so $v(1)=1$, $v(2)=4$, and $d v = 2u d u$. Now solving for $u d u$, we get: $$u d u = \frac{1}{2} d v$$ Substituting $v$ and $d v$ back into the integral, we get: $$\int_{1}^{4} \frac{1}{2(1+v)} d v$$ #Step 4: Evaluate the integral#

Now we have an integral that can be integrated easily. Integrating the function gives: $$\frac{1}{2} \int_{1}^{4} \frac{1}{1+v} d v = \frac{1}{2} [\ln(|1+v|)]_{1}^{4}$$ #Step 5: Applying the fundamental theorem of calculus#

Applying the fundamental theorem of calculus: $$\frac{1}{2} [\ln(|1+v|)]_{1}^{4} = \frac{1}{2}(\ln(5) - \ln(2))$$ #Step 6: Simplify the expression#

Now let's simplify the expression: $$\frac{1}{2}(\ln(5) - \ln(2)) = \frac{1}{2}\ln\left(\frac{5}{2}\right)$$ The final evaluation of the integral is: $$\int_{0}^{\ln 2} \frac{e^{x}}{1+e^{2 x}} d x = \frac{1}{2}\ln\left(\frac{5}{2}\right)$$

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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 technique used to simplify integrals by transforming them into a form that is easier to handle. The basic idea is to pick a substitution that makes the integral more manageable.
In our example, we started by letting $u = e^x$. This substitution simplifies the integral bounds and transforms the expression into a new variable, $u$.
This is helpful because:

It reduces complex expressions into simpler form.
Makes integration straightforward where direct integration seems difficult.

Remember, after performing substitution, don鈥檛 forget to change the limits of integration to match the new variable's perspective. This is crucial in definite integrals to obtain the correct result.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the process of differentiation and integration, providing a way to evaluate definite integrals. It essentially tells us that integration can be "inverted" by differentiation.
The theorem consists of two parts:

The first part identifies the integral of a function as a smooth curve.
The second part allows us to compute the definite integral of a function using its antiderivatives.

The application of the theorem simplifies the task by using antiderivatives to evaluate limits. In our integral, we took the antiderivative of the function, and then applied the boundaries given by the integral to find the solution.

Logarithmic Integration

Logarithmic integration comes into play when dealing with integrals involving logarithmic functions. Typically, it covers integrals where the integrand factor has the form of $\frac{1}{1+v^2}$ or $\frac{1}{x}$, often leading to logarithmic functions upon integration.
In the solved example, after performing substitution, the integrand turned into a form that is recognizable as producing a logarithmic antiderivative. Specifically, the integral $\int \frac{1}{1+v} dv$ results in the natural logarithm $\ln(|1+v|)$.
This shows the power of substitution in converting integrals into forms where known antiderivatives can be easily applied to solve the problem. Understanding when to look for and recognize these forms is key to solving integrals involving logarithmic functions.

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91影视

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

Short Answer

Step by step solution

Let's perform the substitution \(u = e^x\), so \(u(0)=1\), \(u(\ln 2)=2\), and \(d u= e^x d x\). This will make the denominator look like a more recognizable expression: a quadratic form. #Step 2: Rewrite the integral in terms of u#

Applying the fundamental theorem of calculus: $$\frac{1}{2} [\ln(|1+v|)]_{1}^{4} = \frac{1}{2}(\ln(5) - \ln(2))$$ #Step 6: Simplify the expression#

Key Concepts

Substitution Method

Fundamental Theorem of Calculus

Logarithmic Integration

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