Problem 56
Question
In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x}{x^{2}+5} d x $$
Step-by-Step Solution
Verified Answer
\( \frac{1}{2} \ln (x^2 + 5) + C \)
1Step 1: Choose the Method
We need to evaluate the integral \( \int \frac{x}{x^2+5} \, dx \). Observing the integral, we notice that the numerator \(x\) is the derivative of the denominator \(x^2+5\), indicating that substitution is the appropriate method.
2Step 2: Define the Substitution
Let \( u = x^2 + 5 \). This means that \( du = 2x \, dx \) or equivalently \( \frac{1}{2} du = x \, dx \). This substitution simplifies the integral with respect to \(u\).
3Step 3: Substitute and Simplify
Using the substitution \(u = x^2 + 5\) and \( \frac{1}{2} du = x \, dx \), substitute into the integral: \[\int \frac{x}{x^2+5} \, dx = \int \frac{1}{u} \cdot \frac{1}{2} \, du = \frac{1}{2} \int \frac{1}{u} \, du.\]
4Step 4: Integrate with Respect to u
The integral \( \frac{1}{2} \int \frac{1}{u} \, du \) is a standard logarithmic integral. Integrating, we get:\[\frac{1}{2} \ln |u| + C,\]where \(C\) is the constant of integration.
5Step 5: Back Substitute to Original Variable
Substitute back \(u = x^2 + 5\) into the solution to express the result in terms of \(x\):\[\frac{1}{2} \ln |x^2 + 5| + C.\]Since \(x^2 + 5\) is always positive for real \(x\), the absolute value symbols can be omitted yielding:\[\frac{1}{2} \ln (x^2 + 5) + C.\]
Key Concepts
Substitution MethodIntegration by PartsLogarithmic Integral
Substitution Method
The substitution method is a powerful technique for simplifying the process of integration. It works by transforming the original integral into a simpler one through a change of variables.
In essence, substitution can be thought of as the reverse of the chain rule in differentiation. We select a substitution that simplifies the integral, often by letting a part of the integral equal a new variable.
For example, in the given problem, we noticed that the numerator, which was \(x\), resembled the derivative of the denominator \(x^2 + 5\). Hence, we defined our substitution as \(u = x^2 + 5\), which automatically provided the relationship \(du = 2x \, dx\).
This step transformed our integral into a straightforward logarithmic form: \(\frac{1}{2} \int \frac{1}{u} \, du\).
In essence, substitution can be thought of as the reverse of the chain rule in differentiation. We select a substitution that simplifies the integral, often by letting a part of the integral equal a new variable.
For example, in the given problem, we noticed that the numerator, which was \(x\), resembled the derivative of the denominator \(x^2 + 5\). Hence, we defined our substitution as \(u = x^2 + 5\), which automatically provided the relationship \(du = 2x \, dx\).
This step transformed our integral into a straightforward logarithmic form: \(\frac{1}{2} \int \frac{1}{u} \, du\).
- Identify a substitution \(u = g(x)\) where \(g'(x)\) appears in the integrand.
- Rewrite the integrand and differential \(du = g'(x) \, dx\).
- Integrate with respect to \(u\).
- Finally, translate back to the original variable, \(x\).
Integration by Parts
Integration by parts is another technique in the integration toolkit. It is particularly useful for integrals involving products of functions that cannot easily be simplified through substitution.
The general formula for integration by parts is derived from the product rule for differentiation: \[\int u \, dv = uv - \int v \, du\]Here’s how to approach integration by parts:
The general formula for integration by parts is derived from the product rule for differentiation: \[\int u \, dv = uv - \int v \, du\]Here’s how to approach integration by parts:
- Choose \(u\) and \(dv\) from the integrand \(uv\). Ideally, \(u\) should be a function that becomes simpler when differentiated.
- Determine \(du\) by differentiating \(u\) and \(v\) by integrating \(dv\).
- Substitute these into the formula \(\int u \, dv = uv - \int v \, du\).
- Solve the resulting integral.
Logarithmic Integral
The logarithmic integral is a type of integral that involves the natural logarithm function. In simple terms, it refers to integrals of the form \(\int \frac{1}{u} \, du\).
This specific form is ubiquitous in calculus, especially after a successful substitution transforms the integral into this simpler form.
The integration result for this standard form is:
Logarithmic integrals are particularly simple once identified but are crucial in the simplification process when solving complex integrals. Recognizing when an integral can be reduced to a logarithmic form is a skillful technique every student should master.
This specific form is ubiquitous in calculus, especially after a successful substitution transforms the integral into this simpler form.
The integration result for this standard form is:
- \(\int \frac{1}{u} \, du = \ln |u| + C\) where \(C\) is the constant of integration.
- This result is direct and comes from understanding that the derivative of \(\ln |u|\) is \(\frac{1}{u}\).
Logarithmic integrals are particularly simple once identified but are crucial in the simplification process when solving complex integrals. Recognizing when an integral can be reduced to a logarithmic form is a skillful technique every student should master.
Other exercises in this chapter
Problem 55
Complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}-2 x+2} d x $$
View solution Problem 55
In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{e}^{e^{2}} \frac{d x}{x(\ln x)^{2}} $$
View solution Problem 56
Complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}+4 x+5} d x $$
View solution Problem 56
In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{1}^{2} \frac{x d x}{\left(x^{2}+1\right) \ln \left(x^{2}+1\right)} $$
View solution