Integrate Using u-Substitution integral of 28(7x-2)^3 with respect to x
The problem is asking to perform an integration using the u-substitution method. This involves identifying a part of the integrand that can be substituted with a new variable, typically denoted as 'u', which simplifies the integration process. In the expression provided, the goal is to integrate a function of the form 28(7x-2)^3 with respect to x by substituting the inner function 7x-2 as 'u' to make the integral easier to evaluate.
$\int 28 \left(\left(\right. 7 x - 2 \left.\right)\right)^{3} d x$
Answer
Solution:
Step:1
Assign $u = 7x - 2$. Consequently, $du = 7dx$, leading to the expression $dx = \frac{1}{7}du$. Proceed by substituting $u$ and $du$ into the integral.
Step:1.1
Set $u = 7x - 2$ and compute $\frac{du}{dx}$.
Step:1.1.1
Take the derivative of $7x - 2$ with respect to $x$: $\frac{d}{dx}(7x - 2)$.
Step:1.1.2
Apply the Sum Rule in differentiation to find the derivative of $7x - 2$ as $\frac{d}{dx}(7x) + \frac{d}{dx}(-2)$.
Step:1.1.3
Determine $\frac{d}{dx}(7x)$.
Step:1.1.3.1
Given that $7$ is a constant, the derivative of $7x$ is $7\frac{d}{dx}(x)$.
Step:1.1.3.2
Utilize the Power Rule, which states that $\frac{d}{dx}(x^n) = nx^{n-1}$ for $n = 1$, to differentiate $x$.
Step:1.1.3.3
Multiply $7$ by $1$ to get $7$.
Step:1.1.4
Employ the Constant Rule for differentiation.
Step:1.1.4.1
Since $-2$ is a constant, its derivative with respect to $x$ is $0$.
Step:1.1.4.2
Combine $7$ and $0$ to obtain $7$.
Step:1.2
Transform the integral into terms of $u$ and $du$: $\int 4u^3 du$.
Step:2
Extract the constant $4$ from the integral: $4 \int u^3 du$.
Step:3
Apply the Power Rule for integration to find the antiderivative of $u^3$ as $\frac{1}{4}u^4$: $4\left(\frac{1}{4}u^4 + C\right)$.
Step:4
Simplify the expression.
Step:4.1
Express $4\left(\frac{1}{4}u^4 + C\right)$ as $4\left(\frac{1}{4}\right)u^4 + C$.
Step:4.2
Rewrite $4\left(\frac{1}{4}\right)u^4 + C$ as $u^4 + C$.
Step:4.3
Multiply $u^4$ by $1$ to maintain the expression as $u^4 + C$.
Step:5
Substitute back the original variable: $(7x - 2)^4 + C$.
Knowledge Notes:
The u-substitution method is a technique used in calculus to simplify the process of integration, particularly when dealing with composite functions. It involves the following steps:
Choosing u: Identify a part of the integrand that can be substituted with a new variable $u$, which simplifies the integral.
Differentiating u: Compute $\frac{du}{dx}$ to find the relationship between $dx$ and $du$.
Substituting: Replace all instances of the chosen part of the integrand with $u$ and $dx$ with the corresponding $du$ expression.
Integrating: Perform the integration with respect to $u$.
Back-substitution: Replace $u$ with the original expression to return to the variable $x$.
Simplification: Simplify the resulting expression if possible.
The Power Rule for integration states that $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$ for any real number $n \neq -1$, where $C$ is the constant of integration.
The Sum Rule for differentiation states that the derivative of a sum of two functions is the sum of their derivatives: $\frac{d}{dx}(f(x) + g(x)) = \frac{df}{dx} + \frac{dg}{dx}$.
The Constant Rule for differentiation states that the derivative of a constant is zero: $\frac{d}{dx}(c) = 0$ for any constant $c$.
By applying these rules and the process of u-substitution, complex integrals can be evaluated more easily.
