Problem 41
Question
Simplify the integrand before integrating by parts. $$ \int \ln (\sqrt{x}) d x $$
Step-by-Step Solution
Verified Answer
\( \frac{1}{2}(x \ln(x) - x) + C \)
1Step 1: Simplify the Integrand
Notice that the integrand is \( \ln(\sqrt{x}) \). We can simplify this expression by using the property of logarithms that says \( \ln(a^b) = b \ln(a) \). Therefore, \( \ln(\sqrt{x}) = \ln(x^{1/2}) = \frac{1}{2} \ln(x) \). The integral then becomes: \( \int \frac{1}{2} \ln(x) \, dx \).
2Step 2: Apply the Constant Multiple Rule
Using the constant multiple rule, which allows us to factor out constants from the integral, we get: \[ \int \frac{1}{2} \ln(x) \, dx = \frac{1}{2} \int \ln(x) \, dx \].
3Step 3: Use Integration by Parts
Integration by parts is given by \( \int u \, dv = uv - \int v \, du \). Choose \( u = \ln(x) \) and \( dv = dx \). Then, compute \( du = \frac{1}{x} \, dx \) and \( v = x \).
4Step 4: Substitute and Simplify
Substitute into the integration by parts formula: \[ \frac{1}{2} \left( x \ln(x) - \int x \cdot \frac{1}{x} \, dx \right) = \frac{1}{2} \left( x \ln(x) - \int 1 \, dx \right) \].
5Step 5: Integrate the Remaining Part
The integrand is now simply \( \int 1 \, dx \), which evaluates to \( x \). So the integrated expression becomes: \[ \frac{1}{2} \left( x \ln(x) - x \right) \].
6Step 6: Include the Integration Constant
Finally, don't forget to add the constant of integration, \( C \), resulting in the final answer: \[ \frac{1}{2}(x \ln(x) - x) + C \].
Key Concepts
Logarithmic IntegrationSimplifying IntegralsConstant Multiple RuleDefinite and Indicate Integrals
Logarithmic Integration
Logarithmic integration involves integrating functions that have logarithmic expressions.
A common scenario involves simplifying the logarithmic expression before performing the integration, which can make the process significantly easier.
In this exercise, the function is initially given as \( \ln(\sqrt{x}) \). By applying the logarithmic identity \( \ln(a^b) = b \ln(a) \), you can rewrite it as \( \frac{1}{2} \ln(x) \).
This simplification is crucial because it lets us express the logarithm in a more manageable form, reducing the complexity of the integration process.
Understanding and recognizing these properties can make solving integrals involving logarithms more straightforward.
A common scenario involves simplifying the logarithmic expression before performing the integration, which can make the process significantly easier.
In this exercise, the function is initially given as \( \ln(\sqrt{x}) \). By applying the logarithmic identity \( \ln(a^b) = b \ln(a) \), you can rewrite it as \( \frac{1}{2} \ln(x) \).
This simplification is crucial because it lets us express the logarithm in a more manageable form, reducing the complexity of the integration process.
Understanding and recognizing these properties can make solving integrals involving logarithms more straightforward.
Simplifying Integrals
Simplifying integrals can drastically reduce the complexity of solving them.
Instead of dealing with complicated expressions, try to rewrite them using known mathematical identities or properties.
It often leads to a more direct path to the solution, enabling the use of more efficient integration techniques.
Instead of dealing with complicated expressions, try to rewrite them using known mathematical identities or properties.
- In the given example, the integrand was simplified using a logarithmic property.
- The expression \( \int \ln(\sqrt{x}) \, dx \) was simplified to \( \int \frac{1}{2} \ln(x) \, dx \).
It often leads to a more direct path to the solution, enabling the use of more efficient integration techniques.
Constant Multiple Rule
The constant multiple rule is a fundamental integration technique that allows you to pull a constant factor out of an integral.
This can simplify calculations substantially.
For instance, in this problem, once the integrand has been simplified to \( \frac{1}{2} \ln(x) \),
This technique is highly useful in both indefinite and definite integrals, as it works universally regardless of the limits of integration.
This can simplify calculations substantially.
For instance, in this problem, once the integrand has been simplified to \( \frac{1}{2} \ln(x) \),
- The constant \( \frac{1}{2} \) can be factored out to give \( \frac{1}{2} \int \ln(x) \, dx \).
This technique is highly useful in both indefinite and definite integrals, as it works universally regardless of the limits of integration.
Definite and Indicate Integrals
Understanding the difference between definite and indefinite integrals is key to mastering integration.
An indefinite integral is represented without any limits of integration, providing a general solution with an added constant, \( C \).
A definite integral, on the other hand, is evaluated over a specific interval with limits.
An indefinite integral is represented without any limits of integration, providing a general solution with an added constant, \( C \).
A definite integral, on the other hand, is evaluated over a specific interval with limits.
- The result of a definite integral is always a number, representing the area under the curve within the specified limits.
- While an indefinite integral will have the form \( \frac{1}{2}(x \ln(x) - x) + C \), solving a definite integral would replace \( C \) by evaluating the result at its boundaries.
Other exercises in this chapter
Problem 41
Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{\infty} \frac{x}{1+x^{2}} d x $$
View solution Problem 41
Calculate each of the definite integrals. $$ \int_{-1}^{1} \frac{x+6}{(x-2)(x+2)} d x $$
View solution Problem 41
Each of the integrands involves an expression of the form \(a^{2}-b^{2} x^{2}, a^{2}+b^{2} x^{2},\) or \(b^{2} x^{2}-a^{2} .\) Use an indirect substitution of t
View solution Problem 42
Calculate the given integral. \(\int \frac{5 x^{2}-2 x+2}{x^{3}+1} d x\)
View solution