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

Solution:

Step 1:

Decompose the integrand $\frac{1+x}{1+x^2}$ into two separate terms.

$$\int \frac{1}{1+x^2}dx+\int \frac{x}{1+x^2}dx$$

Step 2:

Consider each term as an individual integral.

$$\int \frac{1}{1+x^2}dx+\int \frac{x}{1+x^2}dx$$

Step 3:

Express the constant term $1$ as $1^2$ to match the denominator's format.

$$\int \frac{1}{1^2+x^2}dx+\int \frac{x}{1+x^2}dx$$

Step 4:

Solve the first integral which is a standard arctangent function.

$$\arctan (x)+C+\int \frac{x}{1+x^2}dx$$

Step 5:

Perform a u-substitution for the second integral, where $u=1+x^2$.

Step 5.1:

Define the substitution $u=1+x^2$ and differentiate it with respect to $x$.

Step 5.1.1:

Take the derivative of $1+x^2$.

$$\frac{d}{dx}(1+x^2)$$

Step 5.1.2:

Apply the Sum Rule to find the derivative of the sum.

$$\frac{d}{dx}(1)+\frac{d}{dx}(x^2)$$

Step 5.1.3:

Since the derivative of a constant is zero, the derivative of $1$ is $0$.

$$0+\frac{d}{dx}(x^2)$$

Step 5.1.4:

Use the Power Rule, where the derivative of $x^n$ is $n{x}^{n-1}$ with $n=2$.

$$0+2x$$

Step 5.1.5:

Combine the terms to get the derivative of $u$.

$$2x$$

Step 5.2:

Substitute $u$ and $du$ into the integral.

$$\arctan (x)+C+\int \frac{1}{u}\cdot \frac{1}{2}du$$

Step 6:

Simplify the integral.

Step 6.1:

Combine $\frac{1}{u}$ and $\frac{1}{2}$.

$$\arctan (x)+C+\int \frac{1}{2u}du$$

Step 6.2:

Rearrange the terms to place the constant outside the integral.

$$\arctan (x)+C+\frac{1}{2}\int \frac{1}{u}du$$

Step 7:

Extract the constant $\frac{1}{2}$ from the integral.

$$\arctan (x)+C+\frac{1}{2}\int \frac{1}{u}du$$

Step 8:

Integrate $\frac{1}{u}$ with respect to $u$ to get the natural logarithm.

$$\arctan (x)+C+\frac{1}{2}(\ln (|u|)+C)$$

Step 9:

Combine the constants and simplify the expression.

$$\arctan (x)+\frac{1}{2}\ln (|u|)+C$$

Step 10:

Substitute back the original expression for $u$.

$$\arctan (x)+\frac{1}{2}\ln (|1+x^2|)+C$$

Knowledge Notes:

1. U-Substitution: This is a technique used to simplify the integration process by substituting a part of the integrand with a new variable $u$. This often simplifies the integral into a more recognizable form.

2. Integral of Arctangent: The integral of $\frac{1}{1+x^2}$ with respect to $x$ is the arctangent function, $\arctan (x)+C$.

3. Sum Rule in Differentiation: This rule states that the derivative of a sum of two functions is the sum of their derivatives.

4. Power Rule in Differentiation: This rule states that the derivative of $x^n$ with respect to $x$ is $n{x}^{n-1}$.

5. Natural Logarithm Integral: The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (|u|)+C$, where $C$ is the constant of integration.

6. Constants in Integration: Constants can be moved in and out of the integral sign. This property is often used to simplify the integration process.

7. Absolute Value in Logarithms: When integrating functions that can take on negative values, the absolute value is used in the logarithm to ensure the argument is positive, as the logarithm of a negative number is not defined in the real number system.