Integrate Using u-Substitution integral of x square root of 1-x^2 with respect to x
The problem requests to perform an integration of the given function x√(1-x^2) with respect to x using the u-substitution method. U-substitution is a technique commonly used to simplify integrals by substituting a part of the integrand with a new variable, u, thus transforming the complicated integral into an easier form that can be managed with standard integration techniques. The goal is to provide an explanation of how to apply u-substitution to this integral but not to solve the integral itself.
$\int x \sqrt{1 - x^{2}} d x$
Answer
Solution:
Step:1
Choose $u = 1 - x^{2}$. Then, differentiate to find $du$: $du = -2x dx$, which leads to $-\frac{1}{2} du = x dx$. Now, express the integral in terms of $u$ and $du$.
Step:1.1
Define $u = 1 - x^{2}$ and compute $\frac{du}{dx}$.
Step:1.1.1
Take the derivative of $1 - x^{2}$: $\frac{d}{dx} (1 - x^{2})$.
Step:1.1.2
Apply the derivative operation.
Step:1.1.2.1
Using the Sum Rule, the derivative of $1 - x^{2}$ is the sum of the derivatives $\frac{d}{dx} (1) + \frac{d}{dx} (-x^{2})$.
Step:1.1.2.2
The derivative of a constant is zero: $0 + \frac{d}{dx} (-x^{2})$.
Step:1.1.3
Compute $\frac{d}{dx} (-x^{2})$.
Step:1.1.3.1
The derivative of $-x^{2}$, considering $-1$ as a constant, is $- \frac{d}{dx} (x^{2})$.
Step:1.1.3.2
Apply the Power Rule, which states that the derivative of $x^{n}$ is $nx^{n-1}$, where $n = 2$: $0 - 2x$.
Step:1.1.3.3
Combine the constant with $2$: $0 - 2x$.
Step:1.1.4
Subtract $2x$ from $0$: $-2x$.
Step:1.2
Transform the integral into $u$ and $du$: $\int \sqrt{u} \left(-\frac{1}{2}\right) du$.
Step:2
Begin simplifying the integral.
Step:2.1
Place the negative sign in front of the integral: $\int \sqrt{u} \left(-\frac{1}{2}\right) du$.
Step:2.2
Combine the square root and the fraction: $\int -\frac{\sqrt{u}}{2} du$.
Step:3
Extract the constant $-1$ from the integral: $-\int \frac{\sqrt{u}}{2} du$.
Step:4
Extract the constant $\frac{1}{2}$ from the integral: $-\left(\frac{1}{2} \int \sqrt{u} du\right)$.
Step:5
Express $\sqrt{u}$ as $u^{\frac{1}{2}}$: $-\frac{1}{2} \int u^{\frac{1}{2}} du$.
Step:6
Integrate $u^{\frac{1}{2}}$ using the Power Rule to obtain $\frac{2}{3} u^{\frac{3}{2}}$: $-\frac{1}{2} \left(\frac{2}{3} u^{\frac{3}{2}} + C\right)$.
Step:7
Simplify the expression.
Step:7.1
Combine the constants and the integral result: $-\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$.
Step:7.2
Simplify the fraction: $-\frac{1}{3} u^{\frac{3}{2}} + C$.
Step:8
Substitute back the original $u$ value: $-\frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + C$.
Knowledge Notes:
u-Substitution: A technique used in integration, which involves a change of variable to simplify the integral. It is often used when an integral contains a function and its derivative.
Derivative Rules:
Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
Constant Rule: The derivative of a constant is zero.
Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Integration Rules:
Power Rule for Integration: If $f(x) = x^n$, then $\int f(x) dx = \frac{1}{n+1}x^{n+1} + C$, where $C$ is the constant of integration.
Constants can be factored out of an integral.
Square Root as a Power: The square root of a variable $a$ can be expressed as $a^{1/2}$.
Simplifying Expressions: Combining constants and simplifying fractions are common steps in finding a more compact form of the integral result.
Substituting Back: After integrating with respect to the new variable $u$, it is necessary to substitute back in terms of the original variable to complete the problem.
