Integrate Using u-Substitution integral of (x^3)/(5-1/2x^4) with respect to x

Solution:

Step 1

Assign $u = 5 - \frac{1}{2} x^{4}$. Compute the differential $du$.

Step 1.1

Set $u = 5 - \frac{1}{2} x^{4}$ and differentiate with respect to $x$ to find $du$.

Step 1.1.1

Take the derivative of $5 - \frac{1}{2} x^{4}$ with respect to $x$: $\frac{d}{d x} [5 - \frac{1}{2} x^{4}]$.

Step 1.1.2

Perform the differentiation.

Step 1.1.2.1

Apply the Sum Rule to find the derivative of $5 - \frac{1}{2} x^{4}$: $\frac{d}{d x} [5] + \frac{d}{d x} [- \frac{1}{2} x^{4}]$.

Step 1.1.2.2

The derivative of a constant is zero: $0 + \frac{d}{d x} [- \frac{1}{2} x^{4}]$.

Step 1.1.3

Calculate the derivative $\frac{d}{d x} [- \frac{1}{2} x^{4}]$.

Step 1.1.3.1

Using the constant multiple rule, the derivative of $- \frac{1}{2} x^{4}$ is $- \frac{1}{2} \cdot \frac{d}{d x} [x^{4}]$.

Step 1.1.3.2

Apply the Power Rule, which states that $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ for $n = 4$: $- \frac{1}{2} \cdot (4 x^{3})$.

Step 1.1.3.3

Simplify the constant multiplication: $- 2 \cdot (\frac{1}{2} x^{3})$.

Step 1.1.3.4

Combine the constants: $- 2 \cdot \frac{1}{2} x^{3}$.

Step 1.1.3.5

Simplify the expression: $- 2 \cdot \frac{1}{2} x^{3}$.

Step 1.1.3.6

Simplify by canceling common factors.

Step 1.1.3.6.1

Extract the common factor of 2: $- 2 \cdot \frac{1}{2} x^{3}$.

Step 1.1.3.6.2

Cancel out the common factors.

Step 1.1.3.6.2.1

Factor out 2 from the numerator: $- 2 \cdot \frac{1}{2} x^{3}$.

Step 1.1.3.6.2.2

Eliminate the common factor: $- 2 x^{3}$.

Step 1.1.3.6.2.3

Rewrite the result: $- 2 x^{3}$.

Step 1.1.3.6.2.4

Divide by 1 to simplify: $- 2 x^{3}$.

Step 1.1.4

Subtract the result from zero: $- 2 x^{3}$.

Step 1.2

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

Step 2

Simplify the integral.

Step 2.1

Place the negative sign outside the integral: $- \int \frac{1}{u} \cdot \frac{1}{2} du$.

Step 2.2

Combine the fraction: $- \int \frac{1}{2u} du$.

Step 2.3

Rearrange the constant: $- \int \frac{1}{2u} du$.

Step 3

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

Step 4

Integrate $\frac{1}{u}$ with respect to $u$: $- \frac{1}{2} \ln |u| + C$.

Step 5

Simplify the result: $- \frac{1}{2} \ln |u| + C$.

Step 6

Substitute back the original expression for $u$: $- \frac{1}{2} \ln |5 - \frac{1}{2} x^{4}| + C$.

Knowledge Notes:

The process of solving the integral of a function using u-substitution involves several steps and the application of various calculus rules:

1. u-Substitution: This is a technique used to simplify integrals by substituting part of the integral with a new variable, typically denoted as $u$. The goal is to transform the integral into a simpler form that is easier to evaluate.

2. Differentiation Rules: The solution uses the Sum Rule, which states that the derivative of a sum is the sum of the derivatives, and the Power Rule, which states that the derivative of $x^n$ with respect to $x$ is $n x^{n-1}$.

3. Constant Multiple Rule: This rule is used when taking the derivative of a constant multiplied by a function. The derivative is the constant multiplied by the derivative of the function.

4. Integration: After simplifying the integral using u-substitution, the integral of $\frac{1}{u}$ with respect to $u$ is evaluated. The integral of $\frac{1}{u}$ is $\ln|u|$, where $\ln$ denotes the natural logarithm.

5. Back-Substitution: Once the integral is evaluated in terms of $u$, the original variable $x$ is substituted back in place of $u$ to express the result in terms of the original variable.

6. Constants of Integration: When performing indefinite integration, a constant of integration, typically denoted as $C$, is added to the result to account for the family of antiderivatives.

Throughout the process, algebraic manipulation, such as factoring and canceling common factors, is used to simplify expressions.