Integrate Using u-Substitution integral of (1+x)/(1+x^2) with respect to x
The question is asking for the evaluation of an integral using the method known as u-substitution. This is a technique that simplifies integration by changing the variable of integration to one that makes the integral easier to solve. Specifically, the question is focused on finding the antiderivative of the function (1+x)/(1+x^2) with respect to x, by first substituting a part of the integrand with a new variable u, and then finding the integral with respect to this new variable.
$\int \frac{1 + x}{1 + x^{2}} d x$
Answer
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 $nx^{n-1}$ 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:
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.
Integral of Arctangent: The integral of $\frac{1}{1 + x^2}$ with respect to $x$ is the arctangent function, $\arctan(x) + C$.
Sum Rule in Differentiation: This rule states that the derivative of a sum of two functions is the sum of their derivatives.
Power Rule in Differentiation: This rule states that the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Natural Logarithm Integral: The integral of $\frac{1}{u}$ with respect to $u$ is $\ln(|u|) + C$, where $C$ is the constant of integration.
Constants in Integration: Constants can be moved in and out of the integral sign. This property is often used to simplify the integration process.
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.
