Problem 15
Question
Find an antiderivative. $$ f(x)=5 x-\sqrt{x} $$
Step-by-Step Solution
Verified Answer
The antiderivative is \( F(x) = \frac{5x^2}{2} - \frac{2}{3}x^{3/2} + C \).
1Step 1: Identify the Terms
The function given is \( f(x) = 5x - \sqrt{x} \). It consists of two terms: \( 5x \) and \( \sqrt{x} \). We will find the antiderivative of each term separately.
2Step 2: Antiderivative of 5x
The antiderivative of a term in the form \( ax^n \) is \( \frac{a}{n+1}x^{n+1} \) for \( n eq -1 \). Thus, for the term \( 5x \), we can write it as \( 5x^1 \). Applying the formula, we get:\[ \int 5x \, dx = \frac{5}{2}x^2 = \frac{5x^2}{2} \]
3Step 3: Antiderivative of \( \sqrt{x} \)
The term \( \sqrt{x} \) can be written as \( x^{1/2} \). Using the antiderivative formula, we have:\[ \int x^{1/2} \, dx = \frac{1}{1/2+1}x^{1/2+1} = \frac{2}{3}x^{3/2} \]
4Step 4: Combine the Antiderivatives
Combine the antiderivatives from Steps 2 and 3. Don't forget to add the constant of integration \( C \) to account for any constant that could have been present before taking the derivative:\[ F(x) = \frac{5x^2}{2} - \frac{2}{3}x^{3/2} + C \]
Key Concepts
IntegralsPower RuleConstant of Integration
Integrals
Integrals are a fundamental concept in calculus that are often referred to as the "inverse" of derivatives. Essentially, if a derivative gives you the rate of change of a function, an integral helps you find the original function given its rate of change.
In our problem, finding the antiderivative means calculating the integral of the given function, \( f(x) = 5x - \sqrt{x} \). We are trying to determine a function, whose derivative is \( f(x) \).
To solve it, we found the integral of each term separately. Doing so requires using the rules of integration that simplify the process and ensure accuracy.
In our problem, finding the antiderivative means calculating the integral of the given function, \( f(x) = 5x - \sqrt{x} \). We are trying to determine a function, whose derivative is \( f(x) \).
To solve it, we found the integral of each term separately. Doing so requires using the rules of integration that simplify the process and ensure accuracy.
- Integrals help in finding areas under curves.
- They are used to solve differential equations.
- Essential in calculating quantities like distances and volumes.
Power Rule
The power rule is a basic technique used in calculus for finding the integral of power functions, which are functions of the form \( x^n \). This rule is beneficial when you are dealing with terms like \( 5x \) or \( \sqrt{x} \).
Here's how it works: to find the antiderivative of \( x^n \), integrate by transforming it to \( \frac{1}{n+1}x^{n+1} \). The key requirement is that \( n eq -1 \). This transformation increases the exponent by one and divides by the new exponent.
For example, applying the power rule to \( 5x \), with \( x^1 \), results in \( \frac{5}{2}x^2 \). Similarly, rewriting \( \sqrt{x} \) as \( x^{1/2} \) and using the power rule yields \( \frac{2}{3}x^{3/2} \).
Here's how it works: to find the antiderivative of \( x^n \), integrate by transforming it to \( \frac{1}{n+1}x^{n+1} \). The key requirement is that \( n eq -1 \). This transformation increases the exponent by one and divides by the new exponent.
For example, applying the power rule to \( 5x \), with \( x^1 \), results in \( \frac{5}{2}x^2 \). Similarly, rewriting \( \sqrt{x} \) as \( x^{1/2} \) and using the power rule yields \( \frac{2}{3}x^{3/2} \).
- Helps transform polynomial-like terms quickly.
- Versatile and used frequently throughout calculus.
- Simplifies integration and guides through calculations.
Constant of Integration
When performing indefinite integration, it’s important to remember to add a constant of integration, represented by \( C \). This constant accounts for the fact that when you differentiate a function, any constant differentiates to zero.
Consider that every function whose derivative gives \( f(x) = 5x - \sqrt{x} \) can differ by a constant value. This means there are infinite antiderivatives, each differing by a constant, and \( C \) represents exactly that in your solution. In our problem, after finding the antiderivatives for both terms and combining them, incorporating \( C \) gives:
\[ F(x) = \frac{5x^2}{2} - \frac{2}{3}x^{3/2} + C \]
Consider that every function whose derivative gives \( f(x) = 5x - \sqrt{x} \) can differ by a constant value. This means there are infinite antiderivatives, each differing by a constant, and \( C \) represents exactly that in your solution. In our problem, after finding the antiderivatives for both terms and combining them, incorporating \( C \) gives:
\[ F(x) = \frac{5x^2}{2} - \frac{2}{3}x^{3/2} + C \]
- Each indefinite integral has infinitely many solutions.
- \( C \) ensures completeness of the antiderivative.
- Necessary in solving initial value problems in differential equations.
Other exercises in this chapter
Problem 15
Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{1} e^{-0.2 t} d t $$
View solution Problem 15
Find the integrals in problems. Check your answers by differentiation. $$ \int 12 x^{2} \cos \left(x^{3}\right) d x $$
View solution Problem 16
Use integration by parts twice to evaluate the integral. $$ \int(\ln t)^{2} d t $$
View solution Problem 16
Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{1} 2 e^{x} d x $$
View solution