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Magazine Chapter 5: Problem 9
In Exercises 1 through 20 , find the indicated indefinite integral. \(\int \sqrt{3 x+1} d x\)
Short Answer
Expert verified
\(\frac{2}{9} (3x+1)^{3/2} + C\)
Step by step solution
01
- Choose a substitution
To solve the integral \(\textstyle \int \sqrt{3x+1} \, dx\), start by using a substitution. Let \(u = 3x + 1\).
02
- Differentiate the substitution
Differentiate \(u = 3x + 1\) with respect to \(x\) to find \(du\): \(\frac{du}{dx} = 3\). Rearranging gives \(du = 3 \, dx\), so \(dx = \frac{du}{3}\).
03
- Substitute and transform the integral
Substitute \(u\) and \(dx\) into the original integral. The integral now becomes \(\textstyle \int \sqrt{u} \, \frac{du}{3}\), which simplifies to \(\textstyle \frac{1}{3} \int \sqrt{u} \, du\).
04
- Integrate using the power rule
Rewrite \(\sqrt{u}\) as \(u^{1/2}\). The integral now is \(\frac{1}{3} \int u^{1/2} \, du\). Use the power rule for integration: \(\textstyle \int u^{n} \, du = \frac{u^{n+1}}{n+1}\) where \(n = \frac{1}{2}\). This gives us \(\frac{1}{3} \cdot \frac{u^{3/2}}{3/2}\).
05
- Simplify the result
Simplify \(\frac{1}{3} \cdot \frac{u^{3/2}}{3/2}\): \(\textstyle \frac{1}{3} \cdot \frac{2}{3} \cdot u^{3/2} = \frac{2}{9} u^{3/2} + C\).
06
- Substitute back \(u\) to \(x\)
Finally, substitute \(u = 3x + 1\) back into the expression: \(\frac{2}{9} (3x+1)^{3/2} + C\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
When solving integrals, the **substitution method** can simplify complex expressions. This technique involves changing variables to transform an integral into a simpler form. In our exercise, we used the substitution **u = 3x + 1**.
Here's the step-by-step process:
- **Choose a substitution**: Identify an inner function that, when substituted, makes the integral easier to work with. We chose **u = 3x + 1**.
- **Differentiate the substitution**: Find the derivative of your substitution with respect to x. \(\frac{du}{dx} = 3\) gives us \(du = 3 \, dx\). By rearranging, we get \(dx = \frac{du}{3}\).
- **Transform the integral**: Substitute both the expression for u and dx into the original integral. This changes \(\int \sqrt{3x+1} \, dx\) to \(\int \sqrt{u} \, \frac{du}{3}\), which simplifies further to \(\frac{1}{3} \int \sqrt{u} \, du\).
This substitution helps break complicated integrals into simpler ones that we can solve easily.
Here's the step-by-step process:
- **Choose a substitution**: Identify an inner function that, when substituted, makes the integral easier to work with. We chose **u = 3x + 1**.
- **Differentiate the substitution**: Find the derivative of your substitution with respect to x. \(\frac{du}{dx} = 3\) gives us \(du = 3 \, dx\). By rearranging, we get \(dx = \frac{du}{3}\).
- **Transform the integral**: Substitute both the expression for u and dx into the original integral. This changes \(\int \sqrt{3x+1} \, dx\) to \(\int \sqrt{u} \, \frac{du}{3}\), which simplifies further to \(\frac{1}{3} \int \sqrt{u} \, du\).
This substitution helps break complicated integrals into simpler ones that we can solve easily.
Power Rule for Integration
Another crucial concept in this problem is the **Power Rule for Integration**. This rule is a fundamental technique for finding antiderivatives of power functions. The rule states:
\[\int u^n \, du = \frac{u^{n+1}}{n+1} + C\]
In our problem, after substitution, the integral \(\frac{1}{3} \int \sqrt{u} \, du\) becomes \(\frac{1}{3} \int u^{1/2} \, du\). To solve this using the Power Rule:
- **Rewrite the integrand**: Express the square root as a power, \(u^{1/2}\).
- **Apply the Power Rule**: Integrate using the formula above where **n = 1/2**. This gives us:
\[\int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}\]
- **Simplify**: Multiply by the constant outside the integral, \(\frac{1}{3} \, \frac{2}{3} u^{3/2} = \frac{2}{9} u^{3/2} + C\).
The Power Rule simplifies finding antiderivatives, making integration of power functions straightforward.
\[\int u^n \, du = \frac{u^{n+1}}{n+1} + C\]
In our problem, after substitution, the integral \(\frac{1}{3} \int \sqrt{u} \, du\) becomes \(\frac{1}{3} \int u^{1/2} \, du\). To solve this using the Power Rule:
- **Rewrite the integrand**: Express the square root as a power, \(u^{1/2}\).
- **Apply the Power Rule**: Integrate using the formula above where **n = 1/2**. This gives us:
\[\int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}\]
- **Simplify**: Multiply by the constant outside the integral, \(\frac{1}{3} \, \frac{2}{3} u^{3/2} = \frac{2}{9} u^{3/2} + C\).
The Power Rule simplifies finding antiderivatives, making integration of power functions straightforward.
Simplifying Integrals
The process of **simplifying integrals** is critical for solving them efficiently. This involves making substitutions, applying rules, and reducing the integrand to a manageable form. Our exercise demonstrates several simplification steps:
- **Substitute to simplify**: By letting **u = 3x + 1**, we moved from the original integral to a simpler one involving \(u\). This removed the complex expression inside the square root.
- **Rewrite expressions**: Changing \(\sqrt{u}\) to \(u^{1/2}\) transformed the integral into a form where the Power Rule could easily apply.
- **Factor out constants**: Extract constants from the integral to simplify the expression. In our case, we factored out \(\frac{1}{3}\) from the integrand, making it easier to handle.
- **Simplify the result**: After integrating, simplify the resulting expression. Here we combined \(\frac{1}{3} \, \frac{2}{3} u^{3/2}\) to \(\frac{2}{9} u^{3/2}\).
- **Substitute back**: Finally, revert your substitution for the variable of integration. We substituted **u** back to **3x + 1** at the end.
Through these steps, complex integrals can be broken down into much simpler tasks, making them far more approachable.
- **Substitute to simplify**: By letting **u = 3x + 1**, we moved from the original integral to a simpler one involving \(u\). This removed the complex expression inside the square root.
- **Rewrite expressions**: Changing \(\sqrt{u}\) to \(u^{1/2}\) transformed the integral into a form where the Power Rule could easily apply.
- **Factor out constants**: Extract constants from the integral to simplify the expression. In our case, we factored out \(\frac{1}{3}\) from the integrand, making it easier to handle.
- **Simplify the result**: After integrating, simplify the resulting expression. Here we combined \(\frac{1}{3} \, \frac{2}{3} u^{3/2}\) to \(\frac{2}{9} u^{3/2}\).
- **Substitute back**: Finally, revert your substitution for the variable of integration. We substituted **u** back to **3x + 1** at the end.
Through these steps, complex integrals can be broken down into much simpler tasks, making them far more approachable.
