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

Solution:

Step:1

Choose $u=1-x^2$. Then, differentiate to find $du$: $du=-2xdx$, which leads to $-\frac{1}{2}du=xdx$. 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 $n{x}^{n-1}$, 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 $du$: $\int \sqrt{u}(-\frac{1}{2})du$.

Step:2

Begin simplifying the integral.

Step:2.1

Place the negative sign in front of the integral: $\int \sqrt{u}(-\frac{1}{2})du$.

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: $-(\frac{1}{2}\int \sqrt{u}du)$.

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 $\frac{2}{3}{u}^{\frac{3}{2}}$: $-\frac{1}{2}(\frac{2}{3}{u}^{\frac{3}{2}}+C)$.

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:

1. 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.

2. 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 $f^{\prime }(x)=n{x}^{n-1}$.

3. 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.

4. Square Root as a Power: The square root of a variable $a$ can be expressed as $a^{1/2}$.

5. Simplifying Expressions: Combining constants and simplifying fractions are common steps in finding a more compact form of the integral result.

6. 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.