Integrate Using u-Substitution integral of x/( square root of x^2+1) with respect to x
The problem you've presented is a calculus problem that involves finding an antiderivative, or integral, of a given function by using the method of u-substitution. The function to be integrated is x divided by the square root of (x^2 + 1). U-substitution is a technique that simplifies the process of integrating functions by substituting a part of the integrand with a new variable, typically denoted as 'u'. This often simplifies the integral into a form that is easier to evaluate. The question asks to determine the antiderivative by making an appropriate substitution to integrate the expression with respect to x.
$\int \frac{x}{\sqrt{x^{2} + 1}} d x$
Answer
Solution:
Step 1
Assign $u = x^2 + 1$. Then, calculate $du = 2x dx$, which implies $\frac{1}{2}du = x dx$. 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 $\frac{d}{dx}(x^n) = nx^{n-1}$ 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: $\frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C$.
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:
Choose a substitution that simplifies the integral, typically involving the inner function of a composition of functions.
Differentiate the substitution to find $du$ in terms of $dx$.
Rewrite the integral in terms of the new variable $u$.
Simplify the integral if possible, often by factoring out constants or simplifying algebraic expressions.
Integrate with respect to the new variable $u$.
Simplify the result of the integration if necessary.
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: $\frac{d}{dx}(x^n) = nx^{n-1}$.
The Power Rule of integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$.
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.
