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Magazine Chapter 7: Problem 23
Use integration by substitution and the Fundamental Theorem to evaluate the definite integrals in problem. $$ \int_{0}^{3} \frac{2 x}{x^{2}+1} d x $$
Short Answer
Expert verified
The integral evaluates to \( \ln(10) \).
Step by step solution
01
Identify the Substitution
Start by looking for a substitution that can simplify the integral. Notice that the denominator is \( x^2 + 1 \). A good substitution is \( u = x^2 + 1 \).
02
Differentiate and Express dx in terms of du
Differentiate \( u = x^2 + 1 \) to find \( du \). We get \( du = 2x \, dx \). This implies \( dx = \frac{du}{2x} \).
03
Substitute and Simplify the Integral
Substitute \( u = x^2 + 1 \) and \( dx = \frac{du}{2x} \) into the integral. The integral becomes \( \int \frac{2x}{u} \cdot \frac{du}{2x} = \int \frac{1}{u} \cdot du \). This simplifies to \( \int \frac{1}{u} \, du \).
04
Integrate
Integrate \( \int \frac{1}{u} \, du \) to get \( \ln|u| + C \), where \( C \) is the constant of integration. Since this is a definite integral, we will evaluate without adding \( C \).
05
Change the Limits of Integration
Since we used substitution, we need to change the limits of integration. When \( x = 0 \), \( u = 0^2 + 1 = 1 \). When \( x = 3 \), \( u = 3^2 + 1 = 10 \). So, integrate from 1 to 10.
06
Evaluate the Definite Integral
Evaluate \( \left. \ln |u| \right|_1^{10} \). This means calculate \( \ln(10) - \ln(1) \). Since \( \ln(1) = 0 \), it simplifies to \( \ln(10) \).
07
Write the Final Answer
The final answer to the integral is \( \ln(10) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration by Substitution
Integration by substitution is a clever technique that simplifies complex integrals by transforming them into a simpler form. Think of it as a method that helps us change the variable of integration. When you have an integral that seems too complex, find a substitution that makes it easier. In our example, the integral \[ \int_{0}^{3} \frac{2x}{x^2+1} \, dx \]is simplified by setting \( u = x^2 + 1 \).
- **Why choose** \( u = x^2 + 1 \)? Because its derivative, \( du = 2x \, dx \), mirrors part of the integral鈥攕pecifically, the numerator \( 2x \).
- This turns the initially complicated expression into a simpler form: \( \int \frac{1}{u} \, du \).
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a powerful mathematical principle that connects differentiation and integration. It tells us how to evaluate a definite integral, which represents the accumulation of quantities.
- The theorem states that if a function is continuous over an interval, then the integral of its derivative over that interval is the function's change over that interval.
- In practical terms, the theorem provides a way to compute the area under a curve from one point to another.
Definite Integrals
Definite integrals compute the accumulation of values, such as area under a curve, between two points. Unlike indefinite integrals, which can contain an arbitrary constant of integration, definite integrals calculate a specific value.
- The boundaries of integration, such as from 0 to 3, are crucial components that define the specific intervals to evaluate.
- Once we integrate, we substitute these limits back into the antiderivative equation to find a numerical answer.
