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Chapter 2: Problem 311

Find the definite or indefinite integral. $$ \int_{0}^{1} \frac{d t}{3+2 t} $$

Short Answer

Expert verified
\( \frac{1}{2} \ln \left( \frac{5}{3} \right) \).

Step by step solution

01

Identify the Type of Integral

This is a definite integral because it has specified limits of integration: the lower limit is 0 and the upper limit is 1. The integral to evaluate is \( \int_{0}^{1} \frac{1}{3 + 2t} \, dt \).
02

Simplify the Integrand

Notice that the integrand can be written as a form suitable for substitution. The integrand is \( \frac{1}{3 + 2t} \). This suggests a substitution with a linear function.
03

Choose a Suitable Substitution

Set \( u = 3 + 2t \). Then, differentiate to find \( du \):\[ du = 2 \, dt \]Thus, \( dt = \frac{du}{2} \).
04

Change the Limits of Integration

When \( t = 0 \), \( u = 3 + 2(0) = 3 \). When \( t = 1 \), \( u = 3 + 2(1) = 5 \). So, the new limits of integration are from 3 to 5.
05

Substitute and Integrate

Substitute \( u = 3 + 2t \) and \( dt = \frac{du}{2} \) into the integral:\[ \int_{3}^{5} \frac{1}{u} \cdot \frac{du}{2} = \frac{1}{2} \int_{3}^{5} \frac{1}{u} \, du \]The integral \( \int \frac{1}{u} \, du = \ln |u| \), so:\[ \frac{1}{2} \left[ \ln|u| \right]_{3}^{5} \]
06

Evaluate the Definite Integral

Calculate \( \frac{1}{2} [\ln|5| - \ln|3|] \):\[ \frac{1}{2}( \ln 5 - \ln 3) = \frac{1}{2} \ln \left( \frac{5}{3} \right) \].

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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 useful technique for integrating functions. It involves changing the variable in the integral to make the integration process simpler. In this exercise, the integrand is \( \frac{1}{3 + 2t} \). By recognizing the expression \( 3 + 2t \) as a whole, you can simplify the integration by substituting \( u = 3 + 2t \).
  • First, differentiate the equation \( u = 3 + 2t \) to get \( du = 2 \, dt \).
  • Solve for \( dt \), which gives \( dt = \frac{du}{2} \).
This substitution changes the integrand and makes it easier to integrate, transforming it into an expression in terms of \( u \). Simplifying the integral in terms of \( u \) opens the door to easier calculations and often converts complicated forms into easier ones.
Limits of Integration
When dealing with definite integrals, it's important to adjust the limits of integration properly after substitution. In the original problem, the limits were from 0 to 1.
  • First, calculate the new limits by substituting back into \( u = 3 + 2t \).
  • If \( t = 0 \), then \( u = 3 \); if \( t = 1 \), then \( u = 5 \).
By altering these limits, you ensure that the definite integral remains correct in terms of the new variable \( u \). Every time you change variables in an integral, you should recompute the limits accordingly. This adjustment is crucial to obtaining the right answer, as it accounts for the entire interval originally described.
Logarithmic Integration
Logarithmic integration comes into play when dealing with integrals that resemble the form \( \int \frac{1}{u} \, du \). In this exercise, after substitution, the integral simplifies to \( \frac{1}{2} \int_{3}^{5} \frac{1}{u} \, du \).
  • The integral of \( \frac{1}{u} \) is \( \ln |u| \), a basic rule in calculus.
  • Apply the fundamental theorem of calculus by evaluating this result at the new limits: \([ \ln |u| ]_3^5 \).
This results in \( \ln 5 - \ln 3 \), which simplifies further using properties of logarithms to \( \ln \left( \frac{5}{3} \right) \). The logarithmic form is especially helpful because it allows reverting back easily from a complex ratio to simple arithmetic within the limits of integration. This approach exemplifies not only simplification but leveraging calculus principles effectively.

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Most popular questions from this chapter

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x}, \bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible. [T] Lens: \(y=x^{2}\) and \(y=x\)

Compute \(d y / d x\) by differentiating \(\ln y\). $$ y=x^{-1 / x} $$

Compute \(d y / d x\) by differentiating \(\ln y\). $$ y=e^{-\ln x} $$

For the following exercises, find the antiderivatives for the given functions. $$ \tanh ^{2}(x) \operatorname{sech}^{2}(x) $$

For the following exercises, find the antiderivatives for the given functions. $$ \cosh ^{2}(x) \sinh (x) $$
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