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Magazine Chapter 9: Problem 13
Evaluate the following integrals: \(\int x \sqrt{x+1} d x\)
Short Answer
Expert verified
[\frac{2}{5} (x+1)^{5/2} - \frac{2}{3} (x+1)^{3/2} + C]
Step by step solution
01
Choose a Substitution
To evaluate the integral \(\int x \sqrt{x+1} dx\), choose the substitution \(u = x+1\).
02
Differentiate the Substitution
Differentiate \(u = x+1\) with respect to \(x\). This gives \(du = dx\).
03
Express x in terms of u
Since \(u = x+1\), it follows that \(x = u-1\).
04
Substitute in the Integral
Substitute \(x = u-1\), \(dx = du\), and \(u = x+1\) into the integral. The integral becomes \(\int (u-1) \sqrt{u} du\).
05
Simplify the Integral
Rewrite the integral as \(\int (u-1) \sqrt{u} du = \int u^{3/2} du - \int u^{1/2} du\).
06
Integrate Each Term
Evaluate each term separately: \(\int u^{3/2} du = \frac{2}{5} u^{5/2}\) and \(\int u^{1/2} du = \frac{2}{3} u^{3/2}\).
07
Combine the Results
Combine the results to get \((\frac{2}{5}) u^{5/2} - (\frac{2}{3}) u^{3/2} + C\).
08
Substitute Back u
Substitute \(u = x+1\) back into the expression to get \((\frac{2}{5}) (x+1)^{5/2} - (\frac{2}{3}) (x+1)^{3/2} + C\).
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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 is a powerful tool used in integration to simplify complex integrals. The main idea is to replace a part of the integral with a new variable, usually denoted as \(u\). This helps transform the integral into a more manageable form.
In our example, we used the substitution \(u = x+1\). This allowed us to convert the integral \(\textstyle \int x \sqrt{x+1} dx\) into \(\textstyle \int (u-1) \sqrt{u} du\).
Here's the step-by-step process for using substitution:
In our example, we used the substitution \(u = x+1\). This allowed us to convert the integral \(\textstyle \int x \sqrt{x+1} dx\) into \(\textstyle \int (u-1) \sqrt{u} du\).
Here's the step-by-step process for using substitution:
- Choose a substitution \(u = g(x)\).
- Differentiate to find \(du = g'(x) dx\).
- Express the original \(x\) in terms of \(u\).
- Substitute the expressions for \(x\) and \(dx\) into the integral.
- Simplify the integral and solve it.
- Finally, substitute back the original variable \(x\) into the solution.
Definite Integrals
Definite integrals are used to calculate the area under a curve within a specific interval. They have upper and lower bounds, denoted as \(a\) and \(b\).
A definite integral is written as \(\textstyle \int_a^b f(x) dx\).
The process of evaluating definite integrals involves:
A definite integral is written as \(\textstyle \int_a^b f(x) dx\).
The process of evaluating definite integrals involves:
- Finding the indefinite integral of the function.
- Applying the Fundamental Theorem of Calculus, which states that if \(F(x)\) is the antiderivative of \(f(x)\), then \(\textstyle \int_a^b f(x) dx = F(b) - F(a)\).
Indefinite Integrals
Indefinite integrals represent a family of functions that differ by a constant. They are written without bounds, as \(\textstyle \int f(x) dx\).
The result of an indefinite integral is a function plus a constant of integration, denoted as \(C\).
In our example, we found the indefinite integral of \(\textstyle \int x \sqrt{x+1} dx\).
To solve indefinite integrals:
The result of an indefinite integral is a function plus a constant of integration, denoted as \(C\).
In our example, we found the indefinite integral of \(\textstyle \int x \sqrt{x+1} dx\).
To solve indefinite integrals:
- Identify the function \(f(x)\) to be integrated.
- Choose an appropriate integration technique (e.g., substitution).
- Integrate the function step by step.
- Add the constant of integration \(C\) at the end.
Integration of Polynomial Functions
Polynomial functions can be integrated term-by-term, making them straightforward to handle.
In our exercise, after substitution, the integrand became \(u^{3/2} - u^{1/2}\), which are polynomial terms with fractional exponents.
Here are the steps to integrate polynomial functions:
In our exercise, after substitution, the integrand became \(u^{3/2} - u^{1/2}\), which are polynomial terms with fractional exponents.
Here are the steps to integrate polynomial functions:
- Rewrite the integral in a simplified form, if necessary.
- For each term, use the power rule for integration: \(\textstyle \int u^n du = \frac{u^{n+1}}{n+1} + C\).
- Combine the results.
- Don't forget to substitute back the original variable if you used a substitution.
