Integrate Using u-Substitution integral of 1/((8x-1)^3) with respect to x
The problem provided is a calculus problem specifically asking to perform an integration using the u-substitution method. The integral that needs to be solved is \( \int \frac{1}{(8x-1)^3} \,dx \). The question requires you to use u-substitution, a common technique in integral calculus to simplify integrals and make them easier to evaluate. You would generally choose \( u \) to be a function inside the integral that after differentiation, its derivative or a constant multiple of it can be found in the integral, thereby allowing you to substitute \( du \) for a corresponding expression in terms of \( dx \). The purpose of the question is to find the antiderivative of the given function by transforming the integral into a simpler form through substitution.
$\int \frac{1}{\left(\left(\right. 8 x - 1 \left.\right)\right)^{3}} d x$
Answer
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 $nx^{n-1}$ 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:
$u$-Substitution: This technique requires identifying a part of the integrand that can be substituted with a new variable $u$, simplifying the integral.
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.
Sum Rule: This rule states that the derivative of a sum is the sum of the derivatives.
Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Constant Rule: The derivative of a constant is zero.
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 $n \neq -1$.
Back-Substitution: After integrating with respect to $u$, the original variable $x$ is substituted back into the result to complete the solution.
Simplification: The final step involves simplifying the expression by combining constants and applying basic algebraic rules.
