Problem 3 Express the integral in terms of... [FREE SOLUTION]

91影视

Calculus - AP Edition

Howard Anton, Irl Bivens

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 5: Problem 3

Express the integral in terms of the variable $u$, but do not evaluate it. (a) $\int_{0}^{1} e^{2 x-1} d x ; u=2 x-1$ (b) $\int_{e}^{e^{2}} \frac{\ln x}{x} d x ; u=\ln x$

Short Answer

Expert verified

(a) $\frac{1}{2} \int_{-1}^{1} e^{u} \, du$; (b) $\int_{1}^{2} u \, du$.

Step by step solution

Substitution for Part (a)

Identify the substitution for the integral $ \int_{0}^{1} e^{2x - 1} \, dx $ using $ u = 2x - 1 $. This implies that $ du = 2 \, dx $ or $ dx = \frac{1}{2} \, du $.

Change of Limits for Part (a)

Using the substitution $ u = 2x - 1 $, calculate the new limits of integration. When $ x = 0 $, $ u = 2(0) - 1 = -1 $. When $ x = 1 $, $ u = 2(1) - 1 = 1 $. Thus, the new limits are from $ -1 $ to $ 1 $.

Express the Integral in Terms of $ u $ for Part (a)

Substitute $ u = 2x - 1 $ and $ dx = \frac{1}{2} \, du $ into the integral: \[ \int_{-1}^{1} e^{u} \frac{1}{2} \, du = \frac{1}{2} \int_{-1}^{1} e^{u} \, du \].

Substitution for Part (b)

Identify the substitution for the integral $ \int_{e}^{e^2} \frac{\ln x}{x} \, dx $ using $ u = \ln x $. This implies that $ du = \frac{1}{x} \, dx $, so $ dx = x \, du $.

Change of Limits for Part (b)

Using the substitution $ u = \ln x $, calculate the new limits of integration. When $ x = e $, $ u = \ln(e) = 1 $. When $ x = e^2 $, $ u = \ln(e^2) = 2 $. Thus, the new limits are from $ 1 $ to $ 2 $.

Express the Integral in Terms of $ u $ for Part (b)

Substitute $ u = \ln x $ and $ dx = x \, du $ into the integral. Since $ x = e^u $, the integral becomes \[ \int_{1}^{2} u \, du \].

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Key Concepts

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

Change of Variables

The change of variables, also known as substitution, is a crucial technique in integration. It simplifies a complex integral by converting it into a simpler form. The idea is to replace a variable with a new one, usually denoted as $ u $, which makes the integration process easier. Here's how it works.

Identify a part of the integrand (the function being integrated) that can be substituted with a new variable $ u $.
Express the new variable in terms of the original variable. For example, in part (a) of the exercise, $ u = 2x - 1 $.
Determine $ dx $ in terms of $ du $ using the derivative of $ u $ with respect to $ x $. In our case, for part (a), this gives $ dx = \frac{1}{2} \, du $.

Once the substitution is complete, the integral will have a new expression in terms of $ u $, often making it more straightforward to solve.

Definite Integrals

Definite integrals are used to calculate the exact area under a curve within specified bounds. They are expressed with specified upper and lower limits of integration.

In the context of substitution, definite integrals require us to change the bounds of integration along with the variable.
The integral notation $ \int_{a}^{b} f(x) \, dx $ represents integration of $ f(x) $ from $ x = a $ to $ x = b $.
In the exercise, this is witnessed when transforming the original limits of integration from $ x = 0 $ to $ x = 1 $ into new limits as $ u = -1 $ to $ u = 1 $ in part (a).

This transformation ensures that the limits correspond to the new variable $ u $, effectively adjusting the span of the integral to fit the substituted expression.

Limits of Integration

When performing a change of variables in definite integrals, it's essential to accurately convert the limits of integration. These limits define the bounds over which the integral is calculated.

The original limits of integration with respect to $ x $ must be converted to the corresponding limits in terms of $ u $.
In our example, when $ x = 0 $, substitute into $ u = 2x - 1 $ to find the new lower limit; similarly, when $ x = 1 $, compute the new upper limit.
In part (b), transforming $ x = e $ to $ u = \ln(e) = 1 $ and $ x = e^2 $ to $ u = \ln(e^2) = 2 $ exemplifies this conversion.

This conversion is integral to the substitution method, ensuring the new integral corresponds precisely with the original's intent.

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91影视

Express the integral in terms of the variable \(u\), but do not evaluate it. (a) \(\int_{0}^{1} e^{2 x-1} d x ; u=2 x-1\) (b) \(\int_{e}^{e^{2}} \frac{\ln x}{x} d x ; u=\ln x\)

Short Answer

Step by step solution

Substitution for Part (a)

Change of Limits for Part (a)

Express the Integral in Terms of \( u \) for Part (a)

Substitution for Part (b)

Change of Limits for Part (b)

Express the Integral in Terms of \( u \) for Part (b)

Key Concepts

Change of Variables

Definite Integrals

Limits of Integration

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