Integrate Using u-Substitution integral of 1/((8x-1)^3) with respect to x

Solution:

Step 1 Define $u$ as $8x-1$. Consequently, differentiate $u$ with respect to $x$ to find $du$.

Step 1.1 Set $u=8x-1$ and calculate $\frac{du}{dx}$.

Step 1.1.1 Differentiate $8x-1$ to find $\frac{d}{dx}(8x-1)$.

Step 1.1.2 Apply the Sum Rule to obtain the derivative of $8x-1$ with respect to $x$, which is $\frac{d}{dx}(8x)+\frac{d}{dx}(-1)$.

Step 1.1.3 Determine $\frac{d}{dx}(8x)$.

Step 1.1.3.1 As $8$ is a constant, the derivative of $8x$ with respect to $x$ is $8\frac{d}{dx}(x)$.

Step 1.1.3.2 Use the Power Rule, which states that $\frac{d}{dx}(x^n)$ equals $n{x}^{n-1}$ where $n=1$, to differentiate $x$.

Step 1.1.3.3 Multiply $8$ by $1$ to get $8$.

Step 1.1.4 Apply the Constant Rule for differentiation.

Step 1.1.4.1 Since $-1$ is a constant, its derivative with respect to $x$ is $0$.

Step 1.1.4.2 Combine $8$ and $0$ to obtain $8$.

Step 1.2 Express the integral in terms of $u$ and $du$ as $\int \frac{1}{u^3}\cdot \frac{1}{8}du$.

Step 2 Simplify the integral.

Step 2.1 Combine $\frac{1}{u^3}$ and $\frac{1}{8}$.

Step 2.2 Reposition $8$ to precede $u^3$ in the expression, yielding $\int \frac{1}{8u^3}du$.

Step 3 Extract the constant $\frac{1}{8}$ from the integral, resulting in $\frac{1}{8}\int \frac{1}{u^3}du$.

Step 4 Utilize exponent rules.

Step 4.1 Transform $u^3$ in the denominator to $u^{-3}$.

Step 4.2 Apply exponent multiplication rules.

Step 4.2.1 Invoke the rule $(a^m{)}^n=a^{mn}$.

Step 4.2.2 Multiply $3$ by $-1$ to get $u^{-3}$.

Step 5 Integrate $u^{-3}$ with respect to $u$ using the Power Rule to obtain $-\frac{1}{2}{u}^{-2}$.

Step 6 Further simplify the expression.

Step 6.1 Rewrite $-\frac{1}{2}{u}^{-2}$ as $-\frac{1}{2}\cdot \frac{1}{u^2}$.

Step 6.2 Express the result as $-\frac{1}{2u^2}$.

Step 6.3 Finalize the simplification.

Step 6.3.1 Reposition the $2$ to the left of $u^2$.

Step 6.3.2 Multiply $\frac{1}{8}$ by $-\frac{1}{2u^2}$.

Step 6.3.3 Calculate the product of $2$ and $8$ to get $16$.

Step 7 Substitute $8x-1$ back in place of $u$ to complete the integration, resulting in $-\frac{1}{16(8x-1{)}^2}+C$.

Knowledge Notes:

The problem involves integrating a function using $u$-substitution, a common technique in calculus for simplifying integrals. The process involves several key steps and knowledge points:

1. $u$-Substitution: This technique requires identifying a part of the integrand that can be substituted with a new variable $u$, simplifying the integral.

2. Differentiation: To find $du$, the derivative of $u$ with respect to $x$ is calculated. This involves rules of differentiation such as the Sum Rule, Power Rule, and Constant Rule.

3. Sum Rule: This rule states that the derivative of a sum is the sum of the derivatives.

4. Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $n{x}^{n-1}$.

5. Constant Rule: The derivative of a constant is zero.

6. Integration: After simplifying the integral using $u$-substitution, the integral is evaluated using the Power Rule for integration, which states that $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C$ for any real number $n\ne -1$.

7. Back-Substitution: After integrating with respect to $u$, the original variable $x$ is substituted back into the result to complete the solution.

8. Simplification: The final step involves simplifying the expression by combining constants and applying basic algebraic rules.