Integrate Using u-Substitution integral of 6(1+6x)^4 with respect to x

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 $n{x}^{n-1}$, 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 $n{x}^{n-1}$.

• Integration Rules:

  • Power Rule for Integration: The integral of $u^n$ with respect to $u$ is $\frac{u^{n+1}}{n+1}$, provided that $n\ne -1$.

• 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.