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

Solution:

Step 1

Assign $u=x^2+1$. Then, calculate $du=2xdx$, which implies $\frac{1}{2}du=xdx$. Substitute $u$ and $du$ into the integral.

Step 1.1

Set $u=x^2+1$. Compute $\frac{du}{dx}$.

Step 1.1.1

Take the derivative of $x^2+1$: $\frac{d}{dx}(x^2+1)$.

Step 1.1.2

Apply the Sum Rule to find the derivative: $\frac{d}{dx}(x^2)+\frac{d}{dx}(1)$.

Step 1.1.3

Use the Power Rule, which states $\frac{d}{dx}(x^n)=n{x}^{n-1}$ for $n=2$: $2x+\frac{d}{dx}(1)$.

Step 1.1.4

Since the derivative of a constant is zero: $2x+0$.

Step 1.1.5

Combine the terms: $2x$.

Step 1.2

Express the integral in terms of $u$ and $du$: $\int \frac{1}{\sqrt{u}}\cdot \frac{1}{2}du$.

Step 2

Proceed to simplify the integral.

Step 2.1

Combine $\frac{1}{\sqrt{u}}$ with $\frac{1}{2}$: $\int \frac{1}{2\sqrt{u}}du$.

Step 2.2

Rearrange the terms: $\int \frac{1}{2\sqrt{u}}du$.

Step 3

Extract the constant $\frac{1}{2}$ from the integral: $\frac{1}{2}\int \frac{1}{\sqrt{u}}du$.

Step 4

Apply exponent rules.

Step 4.1

Rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$: $\frac{1}{2}\int u^{-\frac{1}{2}}du$.

Step 4.2

Express the integral with a negative exponent: $\frac{1}{2}\int u^{-\frac{1}{2}}du$.

Step 4.3

Simplify the exponent expression.

Step 4.3.1

Apply the rule $(a^m{)}^n=a^{mn}$: $\frac{1}{2}\int u^{-\frac{1}{2}}du$.

Step 4.3.2

Combine the exponents: $\frac{1}{2}\int u^{-\frac{1}{2}}du$.

Step 4.3.3

Maintain the negative exponent: $\frac{1}{2}\int u^{-\frac{1}{2}}du$.

Step 5

Integrate using the Power Rule: $\frac{1}{2}(2u^{\frac{1}{2}}+C)$.

Step 6

Simplify the expression.

Step 6.1

Combine the constants: $\frac{1}{2}\cdot 2u^{\frac{1}{2}}+C$.

Step 6.2

Simplify to: $u^{\frac{1}{2}}+C$.

Step 6.3

The result is: $u^{\frac{1}{2}}+C$.

Step 7

Substitute back the original variable: $(x^2+1{)}^{\frac{1}{2}}+C$.

Knowledge Notes:

The problem involves integrating a function using u-substitution, which is a technique to simplify integrals by changing the variable of integration. The process includes the following steps:

1. Choose a substitution that simplifies the integral, typically involving the inner function of a composition of functions.

2. Differentiate the substitution to find $du$ in terms of $dx$.

3. Rewrite the integral in terms of the new variable $u$.

4. Simplify the integral if possible, often by factoring out constants or simplifying algebraic expressions.

5. Integrate with respect to the new variable $u$.

6. Simplify the result of the integration if necessary.

7. Substitute back the original variable to express the antiderivative in terms of the original variable.

Relevant rules used in the process include:

• The Sum Rule of differentiation: $\frac{d}{dx}(f(x)+g(x))=\frac{df}{dx}+\frac{dg}{dx}$.

• The Power Rule of differentiation: $\frac{d}{dx}(x^n)=n{x}^{n-1}$.

• The Power Rule of integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$, for $n\ne -1$.

• Basic rules of exponents, such as $\sqrt[n]{a^x}=a^{\frac{x}{n}}$ and $(a^m{)}^n=a^{mn}$.

These rules and techniques are fundamental in calculus and are widely used for solving various types of integrals.