Integrate Using u-Substitution integral of x/(1+x^2) with respect to x
The problem is asking for the calculation of the integral of a rational function x/(1+x^2) with respect to x using the technique of u-substitution. U-substitution is a method for finding integrals that involves substituting a part of the integral with a new variable, u, which simplifies the integral into an easier form. The problem requires identifying a suitable substitution that will allow for the integral to be computed more straightforwardly and then carrying out the change of variables to perform the integration.
$\int \frac{x}{1 + x^{2}} d x$
Answer
Solution:
Step 1:
Choose $u = 1 + x^2$. Consequently, $du = 2x dx$, which implies $\frac{du}{2} = x dx$. Express the integral in terms of $u$ and $du$.
Step 1.1:
Set $u = 1 + x^2$ and calculate $\frac{du}{dx}$.
Step 1.1.1:
Take the derivative of $1 + x^2$. $\frac{d}{dx} [1 + x^2]$
Step 1.1.2:
Apply the Sum Rule to find the derivative of $1 + x^2$ with respect to $x$. $\frac{d}{dx} [1] + \frac{d}{dx} [x^2]$
Step 1.1.3:
The derivative of the constant $1$ is $0$. $0 + \frac{d}{dx} [x^2]$
Step 1.1.4:
Utilize the Power Rule, which states that $\frac{d}{dx} [x^n]$ equals $nx^{n-1}$ for $n = 2$. $0 + 2x$
Step 1.1.5:
Combine $0$ and $2x$. $2x$
Step 1.2:
Reformulate the integral using $u$ and $du$. $\int \frac{1}{u} \cdot \frac{1}{2} du$
Step 2:
Proceed to simplify.
Step 2.1:
Combine $\frac{1}{u}$ and $\frac{1}{2}$. $\int \frac{1}{2u} du$
Step 2.2:
Reposition the constant $2$ outside the integral. $\int \frac{1}{2u} du$
Step 3:
Extract the constant $\frac{1}{2}$ from the integral, as it is not dependent on $u$. $\frac{1}{2} \int \frac{1}{u} du$
Step 4:
Integrate $\frac{1}{u}$ with respect to $u$ to obtain $\ln |u|$. $\frac{1}{2} (\ln |u| + C)$
Step 5:
Condense the expression. $\frac{1}{2} \ln |u| + C$
Step 6:
Substitute back $1 + x^2$ for $u$. $\frac{1}{2} \ln |1 + x^2| + C$
Knowledge Notes:
The problem involves integrating a function using the method of u-substitution. This technique is often employed when an integral contains a function and its derivative. The steps taken to solve this problem are typical of u-substitution:
Choosing u: The first step is to identify a portion of the integrand that can be substituted with a new variable, $u$. This is often a function inside another function or a part of the integrand whose derivative is also present.
Finding du: After choosing $u$, differentiate it with respect to $x$ to find $du$. This allows us to replace $dx$ with $du$ in the integral.
Rewriting the integral: The integral is then rewritten in terms of $u$ and $du$. This often simplifies the integral into a more recognizable form.
Simplifying: Before integrating, simplify the integral as much as possible. This may involve factoring out constants or combining terms.
Integrating: Perform the integration with respect to $u$. For basic functions, this step often involves applying standard integration rules, such as the power rule or the integral of $1/u$ being $\ln |u|$.
Back-substitution: Finally, replace $u$ with the original expression in terms of $x$ to return to the original variable.
The problem also demonstrates the use of the Sum Rule and Power Rule in differentiation:
Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
These rules are fundamental in calculus and are used frequently in various problems involving differentiation and integration.
