Integrate Using u-Substitution integral of x square root of 4-x^2 with respect to x
The question asks for the integration of the function \(x \sqrt{4 - x^2}\) with respect to \(x\) by employing the method of u-substitution. This is a technique used in calculus to simplify the integration process by substituting the original variable \(x\) with another variable \(u\). The problem typically requires finding an appropriate substitution that will transform the integral into one that is easier to evaluate. Once the substitution is made, you would then integrate with respect to the new variable \(u\), and finally, substitute back to the original variable \(x\) to get the answer in terms of \(x\).
$\int x \sqrt{4 - x^{2}} d x$
Answer
Solution:
Step:1
Choose $u = 4 - x^{2}$. Consequently, $d u = - 2 x d x$ leads to $- \frac{1}{2} d u = x d x$. Transition to expressions with $u$ and $d u$.
Step:1.1
Set $u = 4 - x^{2}$ and compute $\frac{d u}{d x}$.
Step:1.1.1
Take the derivative of $4 - x^{2}$: $\frac{d}{d x} [4 - x^{2}]$.
Step:1.1.2
Proceed to differentiate.
Step:1.1.2.1
Apply the Sum Rule: the derivative of $4 - x^{2}$ is the sum of the derivatives of each term: $\frac{d}{d x} [4] + \frac{d}{d x} [- x^{2}]$.
Step:1.1.2.2
The derivative of a constant is zero: $\frac{d}{d x} [4] = 0$, so we have $0 + \frac{d}{d x} [- x^{2}]$.
Step:1.1.3
Calculate $\frac{d}{d x} [- x^{2}]$.
Step:1.1.3.1
The constant factor rule applies to $- x^{2}$, giving $- \frac{d}{d x} [x^{2}]$.
Step:1.1.3.2
Apply the Power Rule, which states that the derivative of $x^{n}$ is $n x^{n - 1}$, where $n = 2$: $0 - (2 x)$.
Step:1.1.3.3
Combine the constant with the derivative: $0 - 2 x$.
Step:1.1.4
Subtract to find the derivative: $- 2 x$.
Step:1.2
Express the integral in terms of $u$ and $d u$: $\int \sqrt{u} \frac{1}{- 2} d u$.
Step:2
Begin simplification.
Step:2.1
Extract the negative sign from the fraction: $\int \sqrt{u} (- \frac{1}{2}) d u$.
Step:2.2
Merge the square root and the fraction: $\int - \frac{\sqrt{u}}{2} d u$.
Step:3
Extract the constant $- 1$ from the integral: $- \int \frac{\sqrt{u}}{2} d u$.
Step:4
Extract the constant $\frac{1}{2}$ from the integral: $- (\frac{1}{2} \int \sqrt{u} d u)$.
Step:5
Express $\sqrt{u}$ as a power of $u$: $- \frac{1}{2} \int u^{\frac{1}{2}} d u$.
Step:6
Integrate using the Power Rule: $- \frac{1}{2} (\frac{2}{3} u^{\frac{3}{2}} + C)$.
Step:7
Further simplification.
Step:7.1
Combine constants and the power of $u$: $- \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$.
Step:7.2
Simplify the expression: $- \frac{1}{3} u^{\frac{3}{2}} + C$.
Step:8
Substitute $u$ back with $4 - x^{2}$: $- \frac{1}{3} ((4 - x^{2}))^{\frac{3}{2}} + C$.
Knowledge Notes:
The problem involves integrating a function that contains a square root of a quadratic expression. To solve this, we use u-substitution, which is a technique for evaluating integrals. The key steps in u-substitution include:
Choosing a substitution that simplifies the integral.
Differentiating the substitution to find $du$ in terms of $dx$.
Rewriting the integral in terms of the new variable $u$.
Simplifying the integral if possible, including factoring out constants.
Integrating with respect to $u$ using known integration rules, such as the Power Rule.
Simplifying the result of the integration.
Substituting back to the original variable if the problem requires it.
The Power Rule for integration states that $\int x^n dx = \frac{1}{n+1} x^{n+1} + C$ for any real number $n \neq -1$, where $C$ is the constant of integration. This rule is used in Step 6 to integrate $u^{\frac{1}{2}}$ with respect to $u$.
The Sum Rule for differentiation states that the derivative of a sum of functions is the sum of their derivatives. This rule is applied in Step 1.1.2.1 to differentiate $4 - x^{2}$.
The Constant Multiple Rule for differentiation states that the derivative of a constant times a function is the constant times the derivative of the function. This is used in Step 1.1.3.1 when differentiating $- x^{2}$.
The Constant Rule for differentiation states that the derivative of a constant is zero, which is used in Step 1.1.2.2 when differentiating the constant $4$.
