Problem 31
Question
In Problems 17-36, use substitution to evaluate each indefinite integral. $$ \int x \sqrt{1+x^{2}} d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral is \( \int x \sqrt{1+x^{2}} \, dx = \frac{1}{3} (1 + x^2)^{3/2} + C \).
1Step 1: Choose the Substitution
To simplify the integral \( \int x \sqrt{1+x^2} \, dx \), we will use the substitution method. Let's choose \( u = 1 + x^2 \). This substitution simplifies the square root present in the integral.
2Step 2: Differentiate the Substitution
Differentiate \( u = 1 + x^2 \) with respect to \( x \) to find \( du \): \( du = 2x \, dx \). This implies \( x \, dx = \frac{1}{2} du \). We'll use this result to replace \( x \, dx \) in the integral.
3Step 3: Substitute into the Integral
Substitute \( u = 1 + x^2 \) and \( x \, dx = \frac{1}{2} du \) into the integral: \[ \int x \sqrt{1+x^2} \, dx = \int \sqrt{u} \cdot \frac{1}{2} du = \frac{1}{2} \int u^{1/2} \, du \].
4Step 4: Integrate with Respect to u
Now integrate \( \frac{1}{2} \int u^{1/2} \, du \): \[ \frac{1}{2} \int u^{1/2} \, du = \frac{1}{2} \cdot \frac{u^{3/2}}{3/2} = \frac{1}{3} u^{3/2} + C \], where \( C \) is the constant of integration.
5Step 5: Substitute Back in Terms of x
Substitute \( u = 1 + x^2 \) back into the integrated result: \( \frac{1}{3} u^{3/2} + C = \frac{1}{3} (1 + x^2)^{3/2} + C \).
6Step 6: State the Final Result
The final indefinite integral of \( \int x \sqrt{1+x^2} \, dx \) is: \[ \int x \sqrt{1+x^{2}} \, dx = \frac{1}{3} (1 + x^2)^{3/2} + C \], where \( C \) is the constant of integration.
Key Concepts
Substitution MethodIntegration TechniquesCalculus Education
Substitution Method
The substitution method is a powerful integration technique often used to simplify integrals with composite functions. It involves substituting a part of the integral with a single variable, which can make the integration process easier. In this specific problem, we selected the substitution \( u = 1 + x^2 \).
This choice is strategic because it simplifies the expression under the square root. When you differentiate \( u \), you get \( du = 2x \, dx \). This tells us that \( x \, dx = \frac{1}{2} \, du \).
By using substitution, we effectively change the variable of integration from \( x \) to \( u \). This converts the original integral into a simpler form: \( \frac{1}{2} \int u^{1/2} \, du \).
Remember, the goal of substitution is to rewrite the integral in terms of \( u \) so that it becomes something standard or easy to solve.
This choice is strategic because it simplifies the expression under the square root. When you differentiate \( u \), you get \( du = 2x \, dx \). This tells us that \( x \, dx = \frac{1}{2} \, du \).
By using substitution, we effectively change the variable of integration from \( x \) to \( u \). This converts the original integral into a simpler form: \( \frac{1}{2} \int u^{1/2} \, du \).
Remember, the goal of substitution is to rewrite the integral in terms of \( u \) so that it becomes something standard or easy to solve.
Integration Techniques
Beyond the substitution method, there are many integration techniques in calculus that cater to solving different types of problems. Each technique has its own applications and strengths depending on the structure of the integral.
Some common techniques include:
When solving integrals, always look for opportunities to transform the function into a simpler or more recognizable form by wisely choosing the appropriate method.
Some common techniques include:
- Integration by parts
- Trigonometric substitution
- Partial fraction decomposition
- Inverse trigonometric derivatives
- Using known antiderivatives
When solving integrals, always look for opportunities to transform the function into a simpler or more recognizable form by wisely choosing the appropriate method.
Calculus Education
Understanding and applying integration techniques like the substitution method plays a crucial role in calculus education. It equips students with the tools to tackle complex integrals that often appear in engineering, physics, and other sciences.
These skills are not only essential for solving academic problems but also in practical applications. Learning how to recognize when and how to apply each technique strengthens problem-solving abilities.
In calculus education, it's important to regularly practice different integration methods to become proficient. Exercises like this one improve conceptual understanding and help students recognize patterns that make integration simpler in real-world scenarios. Encouraging students to explore a variety of techniques will make them confident in tackling diverse problems they might encounter.
These skills are not only essential for solving academic problems but also in practical applications. Learning how to recognize when and how to apply each technique strengthens problem-solving abilities.
In calculus education, it's important to regularly practice different integration methods to become proficient. Exercises like this one improve conceptual understanding and help students recognize patterns that make integration simpler in real-world scenarios. Encouraging students to explore a variety of techniques will make them confident in tackling diverse problems they might encounter.
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