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

Solution:

Step:1

Choose $u = 1 - x^{2}$. Consequently, we have $d u = - 2 x d x$, which implies $- \frac{1}{2} d u = x d x$. Now express the integral in terms of $u$ and $d u$.

Step:1.1

Set $u = 1 - x^{2}$ and compute $\frac{d u}{d x}$.

Step:1.1.1

Take the derivative of $1 - x^{2}$: $\frac{d}{d x} [ 1 - x^{2} ]$.

Step:1.1.2

Proceed with differentiation.

Step:1.1.2.1

Using the Sum Rule, the derivative of $1 - x^{2}$ is the sum of the derivatives of each term: $\frac{d}{d x} [ 1 ] + \frac{d}{d x} [ - x^{2} ]$.

Step:1.1.2.2

The derivative of a constant, $1$, is $0$: $0 + \frac{d}{d x} [ - x^{2} ]$.

Step:1.1.3

Evaluate the derivative of $- x^{2}$: $\frac{d}{d x} [ - x^{2} ]$.

Step:1.1.3.1

Applying the constant multiple rule, the derivative of $- x^{2}$ is $- 1$ times the derivative of $x^{2}$: $0 - \frac{d}{d x} [ x^{2} ]$.

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 $- 1$ with $2$: $0 - 2 x$.

Step:1.1.4

Subtract $2 x$ from $0$: $- 2 x$.

Step:1.2

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

Step:2

Begin simplification.

Step:2.1

Extract the negative sign from the fraction: $\int \frac{1}{\sqrt{u}} \left( - \frac{1}{2} \right) d u$.

Step:2.2

Combine $\frac{1}{\sqrt{u}}$ with $\frac{1}{2}$: $\int - \frac{1}{\sqrt{u} \cdot 2} d u$.

Step:2.3

Rearrange the fraction: $\int - \frac{1}{2 \sqrt{u}} d u$.

Step:3

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

Step:4

Extract the constant $\frac{1}{2}$ from the integral: $- \left( \frac{1}{2} \int \frac{1}{\sqrt{u}} d u \right)$.

Step:5

Apply exponent rules.

Step:5.1

Express $\sqrt{u}$ as $u^{\frac{1}{2}}$: $- \frac{1}{2} \int \frac{1}{u^{\frac{1}{2}}} d u$.

Step:5.2

Rewrite $u^{\frac{1}{2}}$ in the denominator as $u^{-\frac{1}{2}}$: $- \frac{1}{2} \int u^{-\frac{1}{2}} d u$.

Step:5.3

Simplify the exponent.

Step:5.3.1

Apply the power of a power rule: $- \frac{1}{2} \int u^{\frac{1}{2} \cdot -1} d u$.

Step:5.3.2

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

Step:5.3.3

Keep the negative sign outside the fraction: $- \frac{1}{2} \int u^{-\frac{1}{2}} d u$.

Step:6

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

Step:7

Final simplification.

Step:7.1

Multiply $- \frac{1}{2}$ by $2 u^{\frac{1}{2}}$: $- u^{\frac{1}{2}} + C$.

Step:7.2

Combine terms: $- u^{\frac{1}{2}} + C$.

Step:8

Substitute back $u$ with $1 - x^{2}$: $- ((1 - x^{2})^{\frac{1}{2}}) + C$.

Knowledge Notes:

The u-substitution method is a technique used in calculus to simplify the process of integration by substituting a part of the integrand with a new variable, typically denoted as $u$. This method often simplifies the integral into a form that is easier to integrate.

The steps involved in u-substitution generally include:

1. Identifying a part of the integrand to substitute with $u$.

2. Differentiating $u$ with respect to $x$ to find $du/dx$ and solving for $dx$.

3. Substituting $u$ and $dx$ into the integral.

4. Simplifying the integral if possible.

5. Integrating with respect to $u$.

6. Substituting back the original variable $x$ using the initial substitution.

The Power Rule for integration states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$, provided $n \neq -1$. When $n = -1$, the integral is $\ln|x| + C$.

The Sum Rule for differentiation states that the derivative of a sum of functions is the sum of their derivatives.

The Constant Multiple Rule allows us to take constants out of the derivative or integral.

The Power Rule for differentiation states that the derivative of $x^n$ with respect to $x$ is $n x^{n-1}$.

These rules are fundamental to calculus and are used extensively in solving integration and differentiation problems.