Integrate Using u-Substitution integral of 6(1+6x)^4 with respect to x
This question asks for the computation of an integral using the method of u-substitution. U-substitution is a common technique used in calculus to simplify the process of integration, particularly when dealing with composite functions. The function to be integrated, 6(1+6x)^4, is such that it can be mapped to a simpler function by substituting a part of it with a single variable (u), after which the integral can be more easily computed. After the integral is solved with respect to the new variable u, the solution must be converted back to the original variable x. The question does not require the answer but a clarification of the method to be used.
$\int 6 \left(\left(\right. 1 + 6 x \left.\right)\right)^{4} d x$
Answer
Solution:
Step 1:
Choose $u = 1 + 6x$. Consequently, we have $du = 6dx$, which implies $\frac{du}{6} = dx$. We'll use $u$ and $du$ to reformulate the integral.
Step 1.1:
Set $u = 1 + 6x$ and compute $\frac{du}{dx}$.
Step 1.1.1:
Take the derivative of $1 + 6x$. $\frac{d}{dx}(1 + 6x)$
Step 1.1.2:
Apply differentiation.
Step 1.1.2.1:
Using the Sum Rule, the derivative of $1 + 6x$ with respect to $x$ is $\frac{d}{dx}(1) + \frac{d}{dx}(6x)$.
Step 1.1.2.2:
Given that $1$ is a constant, its derivative with respect to $x$ is zero. $0 + \frac{d}{dx}(6x)$
Step 1.1.3:
Calculate $\frac{d}{dx}(6x)$.
Step 1.1.3.1:
As $6$ is a constant, the derivative of $6x$ with respect to $x$ is $6\frac{d}{dx}(x)$.
Step 1.1.3.2:
Differentiate using the Power Rule, which states that the derivative of $x^n$ is $nx^{n-1}$, where $n = 1$. $0 + 6 \cdot 1$
Step 1.1.3.3:
Multiply $6$ by $1$. $0 + 6$
Step 1.1.4:
Combine $0$ and $6$. $6$
Step 1.2:
Express the integral in terms of $u$ and $du$. $\int u^4 du$
Step 2:
Utilize the Power Rule for integration, which states that the integral of $u^n$ with respect to $u$ is $\frac{u^{n+1}}{n+1}$, hence $\int u^4 du$ becomes $\frac{1}{5}u^5$.
Step 3:
Substitute back $1 + 6x$ for $u$. $\frac{1}{5}(1 + 6x)^5 + C$
Knowledge Notes:
u-Substitution: A technique used in integration, which involves changing the variable of integration to simplify the integral. It is particularly useful when dealing with composite functions.
Derivative Rules:
Sum Rule: The derivative of a sum is the sum of the derivatives.
Constant Rule: The derivative of a constant is zero.
Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Integration Rules:
- Power Rule for Integration: The integral of $u^n$ with respect to $u$ is $\frac{u^{n+1}}{n+1}$, provided that $n \neq -1$.
Constants of Integration: When performing indefinite integration, a constant of integration, typically denoted by $C$, is added to the result to indicate that there are infinitely many antiderivatives.
Back-Substitution: After integrating with respect to $u$, the original variable (in this case, $x$) is substituted back into the expression to return to the original variable of integration.
