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

Solution:

Step:1 Choose $u=x+1$. Consequently, $du=dx$. Reformulate the integral in terms of $u$ and $du$.

Step:1.1 Set $u=x+1$ and compute $\frac{du}{dx}$.

Step:1.1.1 Calculate the derivative of $x+1$. $\frac{d}{dx}[x+1]$

Step:1.1.2 Utilize the Sum Rule to differentiate $x+1$ with respect to $x$: $\frac{d}{dx}[x]+\frac{d}{dx}[1]$.

Step:1.1.3 Apply the Power Rule, which states $\frac{d}{dx}[x^n]=n{x}^{n-1}$ where $n=1$: $1+\frac{d}{dx}[1]$.

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 $du$: $\int \frac{u-1}{u}du$.

Step:2 Decompose the fraction into individual terms: $\int \frac{u}{u}-\frac{1}{u}du$.

Step:3 Separate the integral into two parts: $\int \frac{u}{u}du-\int \frac{1}{u}du$.

Step:4 Proceed to simplify.

Step:4.1 Eliminate the common $u$ terms.

Step:4.1.1 Remove the common factors: $\int \frac{\cancel{u}}{\cancel{u}}du-\int \frac{1}{u}du$.

Step:4.1.2 Restate the expression: $\int du-\int \frac{1}{u}du$.

Step:4.2 Extract the negative sign from the integral: $\int du-\int -\frac{1}{u}du$.

Step:5 Apply the integral of a constant: $u+C-\int -\frac{1}{u}du$.

Step:6 Factor out the constant $-1$: $u+C-(-1)\int \frac{1}{u}du$.

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:

1. Choosing a substitution $u=g(x)$.

2. Computing the derivative $\frac{du}{dx}$.

3. Expressing the integral in terms of $u$.

4. Integrating with respect to $u$.

5. Simplifying the result if possible.

6. Substituting back the original variable $x$.

The Sum Rule states that the derivative of a sum of functions is the sum of their derivatives: $\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$.

The Power Rule is used to differentiate functions of the form $x^n$ and states that $\frac{d}{dx}{x}^n=n{x}^{n-1}$.

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.