Integrate Using u-Substitution integral of (x+3)/(x^2+6x) with respect to x
The problem is asking you to evaluate the definite or indefinite integral of the given function (x + 3)/(x^2 + 6x) with respect to the variable x by employing the u-substitution method. U-substitution is a technique used in calculus to simplify the process of integration by substituting a part of the integral with a new variable, typically denoted as u. This substitution often turns a difficult integral into a simpler one that is easier to evaluate. In this specific integral, you would look for a substitution for u that simplifies the integral of the rational function, which typically involves choosing u so that its derivative du is also present in the integral.
$\int \frac{x + 3}{x^{2} + 6 x} d x$
Answer
Solution:
Step:1
Choose $u = x^{2} + 6x$. Consequently, $du = (2x + 6)dx$, which implies $\frac{du}{2} = (x + 3)dx$. Express the integral in terms of $u$ and $du$.
Step:1.1
Set $u = x^{2} + 6x$ and compute $\frac{du}{dx}$.
Step:1.1.1
Take the derivative of $x^{2} + 6x$: $\frac{d}{dx}(x^{2} + 6x)$.
Step:1.1.2
Apply differentiation rules.
Step:1.1.2.1
Use the Sum Rule to differentiate $x^{2} + 6x$: $\frac{d}{dx}(x^{2}) + \frac{d}{dx}(6x)$.
Step:1.1.2.2
Apply the Power Rule, where $\frac{d}{dx}(x^{n}) = nx^{n-1}$ for $n=2$: $2x + \frac{d}{dx}(6x)$.
Step:1.1.3
Find the derivative of $6x$: $\frac{d}{dx}(6x)$.
Step:1.1.3.1
Since 6 is a constant, its derivative with respect to $x$ is $6\frac{d}{dx}(x)$: $2x + 6\frac{d}{dx}(x)$.
Step:1.1.3.2
Use the Power Rule for $n=1$: $2x + 6 \cdot 1$.
Step:1.1.3.3
Multiply 6 by 1: $2x + 6$.
Step:1.2
Transform the integral into terms of $u$ and $du$: $\int \frac{1}{u} \cdot \frac{1}{2} du$.
Step:2
Simplify the expression.
Step:2.1
Combine $\frac{1}{u}$ and $\frac{1}{2}$: $\int \frac{1}{2u} du$.
Step:2.2
Reposition the constant 2: $\int \frac{1}{2u} du$.
Step:3
Extract the constant $\frac{1}{2}$ from the integral: $\frac{1}{2} \int \frac{1}{u} du$.
Step:4
Integrate $\frac{1}{u}$ with respect to $u$: $\frac{1}{2}(\ln |u| + C)$.
Step:5
Simplify the result: $\frac{1}{2} \ln |u| + C$.
Step:6
Substitute back the original expression for $u$: $\frac{1}{2} \ln |x^{2} + 6x| + C$.
Knowledge Notes:
The problem involves integrating a rational function using the method of u-substitution. This technique is often used when an integral contains a function and its derivative. The steps to solve the problem using u-substitution typically include:
Choosing a substitution: Identify a part of the integral to be $u$, which simplifies the integral when replaced.
Differentiating $u$: Find $du$ by differentiating $u$ with respect to $x$.
Rewriting the integral: Express the original integral in terms of $u$ and $du$.
Simplifying: Simplify the integral if possible, often by factoring out constants or combining like terms.
Integrating with respect to $u$: Perform the integration with the new variable $u$.
Back-substitution: Replace $u$ with the original expression in terms of $x$ to get the final answer.
Relevant rules used in the solution include:
Sum Rule: The derivative of a sum of functions is the sum of the derivatives of those functions.
Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Integration of $\frac{1}{u}$: The integral of $\frac{1}{u}$ with respect to $u$ is $\ln|u|$ plus the constant of integration $C$.
The final result of the integration is expressed in terms of the natural logarithm function, which is the inverse of the exponential function. The absolute value is used in the logarithm to ensure the argument is positive, as the logarithm is only defined for positive real numbers.
