Integrate Using u-Substitution integral of x/(x+1) with respect to x
The question is asking you to perform an integration using a method called u-substitution. In u-substitution, you choose a part of the integrand (the function you are integrating) to be a new variable u, which simplifies the integral into a form that is easier to integrate. In this particular case, you are asked to integrate the function x/(x+1) with respect to x. You would need to identify an appropriate substitution for u, replace x and dx in terms of u and du, and then integrate the transformed expression with respect to u. After integrating, you would then substitute back in terms of x to obtain the final result.
$\int \frac{x}{x + 1} d x$
Answer
Solution:
Step:1 Choose $u = x + 1$. Consequently, $d u = d x$. Reformulate the integral in terms of $u$ and $d u$.
Step:1.1 Set $u = x + 1$ and compute $\frac{d u}{d x}$.
Step:1.1.1 Calculate the derivative of $x + 1$. $\frac{d}{d x} [ x + 1 ]$
Step:1.1.2 Utilize the Sum Rule to differentiate $x + 1$ with respect to $x$: $\frac{d}{d x} [ x ] + \frac{d}{d x} [ 1 ]$.
Step:1.1.3 Apply the Power Rule, which states $\frac{d}{d x} [ x^{n} ] = n x^{n - 1}$ where $n = 1$: $1 + \frac{d}{d x} [ 1 ]$.
Step:1.1.4 Since the derivative of a constant is zero, the derivative of $1$ with respect to $x$ is $0$: $1 + 0$.
Step:1.1.5 Combine $1$ and $0$: $1$.
Step:1.2 Express the integral using $u$ and $d u$: $\int \frac{u - 1}{u} d u$.
Step:2 Decompose the fraction into individual terms: $\int \frac{u}{u} - \frac{1}{u} d u$.
Step:3 Separate the integral into two parts: $\int \frac{u}{u} d u - \int \frac{1}{u} d u$.
Step:4 Proceed to simplify.
Step:4.1 Eliminate the common $u$ terms.
Step:4.1.1 Remove the common factors: $\int \frac{\cancel{u}}{\cancel{u}} d u - \int \frac{1}{u} d u$.
Step:4.1.2 Restate the expression: $\int d u - \int \frac{1}{u} d u$.
Step:4.2 Extract the negative sign from the integral: $\int d u - \int -\frac{1}{u} d u$.
Step:5 Apply the integral of a constant: $u + C - \int -\frac{1}{u} d u$.
Step:6 Factor out the constant $-1$: $u + C - (-1) \int \frac{1}{u} d u$.
Step:7 Integrate $\frac{1}{u}$ with respect to $u$: $u + C - (-\ln |u| + C)$.
Step:8 Combine terms: $u - \ln |u| + C$.
Step:9 Substitute back $u = x + 1$: $x + 1 - \ln |x + 1| + C$.
Knowledge Notes:
The u-substitution method is a technique used in calculus to simplify the process of integration. It involves choosing a new variable $u$ that is a function of $x$, which simplifies the integral. The steps typically involve:
Choosing a substitution $u = g(x)$.
Computing the derivative $\frac{d u}{d x}$.
Expressing the integral in terms of $u$.
Integrating with respect to $u$.
Simplifying the result if possible.
Substituting back the original variable $x$.
The Sum Rule states that the derivative of a sum of functions is the sum of their derivatives: $\frac{d}{d x}(f(x) + g(x)) = \frac{d}{d x}f(x) + \frac{d}{d x}g(x)$.
The Power Rule is used to differentiate functions of the form $x^n$ and states that $\frac{d}{d x}x^n = nx^{n-1}$.
The integral of a constant $a$ with respect to $x$ is $ax + C$, where $C$ is the constant of integration.
The integral of $\frac{1}{u}$ with respect to $u$ is $\ln|u| + C$, where $\ln$ denotes the natural logarithm.
When simplifying integrals, common factors in numerators and denominators can be canceled out, and constants can be factored out of integrals.
