Integrate Using u-Substitution integral of 28(7x-2)^3 with respect to x

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 $\frac{d}{dx}(x^n)=n{x}^{n-1}$ 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 $\frac{1}{4}{u}^4$: $4(\frac{1}{4}{u}^4+C)$.

Step:4

Simplify the expression.

Step:4.1

Express $4(\frac{1}{4}{u}^4+C)$ 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:

1. Choosing u: Identify a part of the integrand that can be substituted with a new variable $u$, which simplifies the integral.

2. Differentiating u: Compute $\frac{du}{dx}$ to find the relationship between $dx$ and $du$.

3. Substituting: Replace all instances of the chosen part of the integrand with $u$ and $dx$ with the corresponding $du$ expression.

4. Integrating: Perform the integration with respect to $u$.

5. Back-substitution: Replace $u$ with the original expression to return to the variable $x$.

6. 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 $n\ne -1$, 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.