Integrate Using u-Substitution integral of x/(1+x^2) with respect to x

Solution:

Step 1:

Choose $u=1+x^2$. Consequently, $du=2xdx$, which implies $\frac{du}{2}=xdx$. 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 $n{x}^{n-1}$ 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:

1. 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.

2. 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.

3. Rewriting the integral: The integral is then rewritten in terms of $u$ and $du$. This often simplifies the integral into a more recognizable form.

4. Simplifying: Before integrating, simplify the integral as much as possible. This may involve factoring out constants or combining terms.

5. 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 $\ln |u|$.

6. 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 $n{x}^{n-1}$.

These rules are fundamental in calculus and are used frequently in various problems involving differentiation and integration.