Problem 16
Question
Use the table of integrals at the back of the book to evaluate the integrals. \(\int q^{2} \sqrt{25-q^{2}} d q\)
Step-by-Step Solution
Verified Answer
\( \frac{125}{3} \sin^{-1}\left(\frac{q}{5}\right) - \frac{q}{2} \sqrt{25-q^{2}} - \frac{25}{2}q + C \).
1Step 1: Identify the Integral Form
The given integral is \( \int q^{2} \sqrt{25-q^{2}} \, dq \). First, identify which standard integral form it resembles from the integral table. This expression is similar to the form \( \int x^{2} \sqrt{a^{2} - x^{2}} \, dx \).
2Step 2: Match Constants and Variables
In our integral, the expression under the square root is \( 25 - q^{2} \), which suggests that \( a^{2} = 25 \) and thus \( a = 5 \). Therefore, \( x \) corresponds to \( q \).
3Step 3: Apply the Standard Integral Formula
According to the table of integrals, the formula for \( \int x^{2} \sqrt{a^{2} - x^{2}} \, dx \) is \( \frac{a^{3}}{3} \sin^{-1}\left(\frac{x}{a}\right) - \frac{x}{2}(a^{2} - x^{2})^{1/2} - \frac{a^{2}}{2}x \). Substituting \( a = 5 \) and \( x = q \) into this formula, we get: \[ \frac{5^{3}}{3} \sin^{-1}\left(\frac{q}{5}\right) - \frac{q}{2}(25 - q^{2})^{1/2} - \frac{25}{2}q \].
4Step 4: Simplify the Expression
Calculate \( \frac{5^{3}}{3} = \frac{125}{3} \). The expression then simplifies to: \[ \frac{125}{3} \sin^{-1}\left(\frac{q}{5}\right) - \frac{q}{2}\sqrt{25 - q^{2}} - \frac{25}{2}q \].
5Step 5: Write the Final Answer
The evaluated integral is: \[ \frac{125}{3} \sin^{-1}\left(\frac{q}{5}\right) - \frac{q}{2}\sqrt{25 - q^{2}} - \frac{25}{2}q + C \], where \( C \) is the constant of integration.
Key Concepts
Integral TableIntegration TechniquesEvaluating Integrals
Integral Table
The integral table is a powerful resource for students and mathematicians alike. It's essentially a compilation of solutions to standard integrals that you frequently encounter in calculus. Rather than performing lengthy calculations each time, you can use the table to quickly find the solution.
In our exercise, we used the integral table to identify the standard form for the given integral \( \int q^{2} \sqrt{25-q^{2}} \, dq \). By matching the integral to the form \( \int x^{2} \sqrt{a^{2} - x^{2}} \, dx \), we saved time and effort. This table makes solving integrals more efficient, as it often provides the final solution based on patterns and known forms.
In our exercise, we used the integral table to identify the standard form for the given integral \( \int q^{2} \sqrt{25-q^{2}} \, dq \). By matching the integral to the form \( \int x^{2} \sqrt{a^{2} - x^{2}} \, dx \), we saved time and effort. This table makes solving integrals more efficient, as it often provides the final solution based on patterns and known forms.
- Understand the structure of the integral table.
- Find the pattern that matches your integral.
- Use substitutions to match variables and constants.
Integration Techniques
Integration techniques are strategies that help solve integrals that are not immediately solvable using basic formulas. In our problem, we used a combination of substitution and pattern matching.
The technique involved identifying the known formula from the integral table and matching it with our exercise. We assigned \( a = 5 \) to fit the pattern \( a^{2} = 25 \). We also equated \( x \) to \( q \) since they play the same role in the integral's structure.
The technique involved identifying the known formula from the integral table and matching it with our exercise. We assigned \( a = 5 \) to fit the pattern \( a^{2} = 25 \). We also equated \( x \) to \( q \) since they play the same role in the integral's structure.
- Pattern matching helps align integrals with known formulas.
- Substitution allows you to adjust variables and constants to fit patterns.
- This method reduces complex integrals into simpler known forms.
Evaluating Integrals
Evaluating integrals involves finding the function that describes the area under a curve. In this exercise, once we obtained the matching formula from the integral table, we substituted \( a \) and \( x \) accordingly.
After substituting \( a = 5 \) and \( x = q \), we applied the formula to compute the integral as \[ \frac{125}{3} \sin^{-1}\left(\frac{q}{5}\right) - \frac{q}{2}\sqrt{25 - q^{2}} - \frac{25}{2}q + C \]. This process provided us with the evaluated integral, representing the original expression in a solved form.
After substituting \( a = 5 \) and \( x = q \), we applied the formula to compute the integral as \[ \frac{125}{3} \sin^{-1}\left(\frac{q}{5}\right) - \frac{q}{2}\sqrt{25 - q^{2}} - \frac{25}{2}q + C \]. This process provided us with the evaluated integral, representing the original expression in a solved form.
- Identify the correct formula from the table.
- Adjust constants and variables by substitution.
- Simplify your expression after applying the formula.
Other exercises in this chapter
Problem 16
Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{0}^{2} \frac{s+1}{\sqrt{4-s^{2}}} d s $$
View solution Problem 16
Evaluate the integrals in Exercises \(15-22\). $$ \int_{0}^{\pi} \sqrt{1-\cos 2 x} d x $$
View solution Problem 16
Evaluate the integrals. \(\int t^{2} e^{4 t} d t\)
View solution Problem 16
In Exercises \(9-16,\) express the integrands as a sum of partial fractions and evaluate the integrals. $$ \int \frac{x+3}{2 x^{3}-8 x} d x $$
View solution