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Magazine Chapter 4: Problem 101
Evaluate. $$ \int x^{2} \sqrt{x^{3}+1} d x $$
Short Answer
Expert verified
\( \int x^2 \sqrt{x^3+1} \, dx = \frac{2}{9} (x^3+1)^{3/2} + C \)
Step by step solution
01
Choose the Substitution
To solve the integral, we will use the substitution method. We choose to let \( u = x^3 + 1 \). The reason for this choice is because the derivative of \( u \) with respect to \( x \) will help simplify the integral involving \( x^2 \).
02
Differentiate the Substitution
Calculate the derivative to find \( du \). Since \( u = x^3 + 1 \), differentiate both sides with respect to \( x \):\[ \frac{du}{dx} = 3x^2 \]This implies that \( du = 3x^2 \, dx \) so \( x^2 \, dx = \frac{1}{3} du \).
03
Substitute in the Integral
Substitute the expressions for \( u \) and \( x^2 \, dx \) into the integral:\[ \int x^2 \sqrt{x^3 + 1} \, dx = \int \, \sqrt{u} \, \frac{1}{3} \, du \]Simplify it:\[ \frac{1}{3} \int \sqrt{u} \, du \]
04
Integrate with Respect to u
Now, integrate \( \frac{1}{3} \int u^{1/2} \, du \):Use the power rule for integration, which states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \):\[ \frac{1}{3} \times \frac{u^{3/2}}{3/2} = \frac{1}{3} \times \frac{2}{3} u^{3/2} = \frac{2}{9} u^{3/2} + C \]
05
Back-Substitute for x
Substitute \( u = x^3 + 1 \) back into the expression:\[ \frac{2}{9} (x^3 + 1)^{3/2} + C \]
06
Write the Final Answer
The evaluated integral is:\[ \int x^2 \sqrt{x^3 + 1} \, dx = \frac{2}{9} (x^3 + 1)^{3/2} + C \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method in integral calculus is like a change of variables that makes integration easier. Think of it as simplifying the problem. In our exercise, we're dealing with the integral \( \int x^2 \sqrt{x^3 + 1} \, dx \). By choosing an appropriate substitution, such as \( u = x^3 + 1 \), we make the expression under the square root simpler.
Here鈥檚 how substitution works step-by-step:
Here鈥檚 how substitution works step-by-step:
- First, identify a function inside the integral that, if replaced, will simplify the expression. Often it鈥檚 something inside a parenthesis or under a square root.
- Check if the derivative of this function exists elsewhere in the integral. In our case, the derivative of \( u = x^3 + 1 \) is \( 3x^2 \), which, after adjusting factors, matches \( x^2 \, dx \) in the original integral.
Power Rule for Integration
Once the integral is expressed in terms of \( u \), the power rule for integration assists us. The power rule is straightforward: for any real number \( n \), \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), provided \( n eq -1 \).
In our problem, we found the integral form \( \frac{1}{3} \int u^{1/2} \, du \). Applying the power rule here simplifies the integral:
In our problem, we found the integral form \( \frac{1}{3} \int u^{1/2} \, du \). Applying the power rule here simplifies the integral:
- Identify \( n \). For \( u^{1/2} \), \( n = 1/2 \).
- Apply the formula: \( \frac{1}{3} \times \frac{u^{3/2}}{3/2} = \frac{2}{9} u^{3/2} \).
Back-Substitution
After integrating with respect to the new variable \( u \), it's time to revert back to the original variable \( x \) to express the final answer in terms of \( x \). This is the "back-substitution" step.
Here's how to do it:
Here's how to do it:
- Recall your substitution: \( u = x^3 + 1 \).
- Replace \( u \) in your integrated function with \( x^3 + 1 \) to convert back: \( \frac{2}{9} (x^3 + 1)^{3/2} + C \).
