Integrate Using u-Substitution integral of 4(6x+1)^5 with respect to x

Solution:

Step:1

Choose $u=6x+1$. Then differentiate to find $du$ in terms of $dx$.

Step:1.1

Set $u=6x+1$ and calculate $\frac{du}{dx}$.

Step:1.1.1

Differentiate $6x+1$ to find $\frac{d}{dx}(6x+1)$.

Step:1.1.2

Apply the Sum Rule to differentiate $6x+1$ as $\frac{d}{dx}(6x)+\frac{d}{dx}(1)$.

Step:1.1.3

Compute $\frac{d}{dx}(6x)$.

Step:1.1.3.1

Since $6$ is a constant, use the constant multiple rule to find $6\frac{d}{dx}(x)$.

Step:1.1.3.2

Apply the Power Rule, which states $\frac{d}{dx}(x^n)=n{x}^{n-1}$ where $n=1$, to get $6\cdot 1$.

Step:1.1.3.3

Multiply $6$ by $1$ to obtain $6$.

Step:1.1.4

Use the Constant Rule for differentiation.

Step:1.1.4.1

The derivative of a constant, $1$, with respect to $x$ is $0$.

Step:1.1.4.2

Combine $6$ and $0$ to get $6$.

Step:1.2

Express the integral in terms of $u$ and $du$ as $\int 4u^5\frac{1}{6}du$.

Step:2

Simplify the integral.

Step:2.1

Combine the constants $\frac{1}{6}$ and $4$.

Step:2.2

Integrate $\frac{4u^5}{6}du$.

Step:2.3

Reduce the fraction by canceling common factors.

Step:2.3.1

Factor out $2$ from $4u^5$.

Step:2.3.2

Remove common factors.

Step:2.3.2.1

Factor $2$ out of $6$.

Step:2.3.2.2

Cancel the $2$'s.

Step:2.3.2.3

Rewrite the integral as $\int \frac{2u^5}{3}du$.

Step:3

Extract the constant $\frac{2}{3}$ from the integral.

Step:4

Apply the Power Rule to integrate $u^5$ with respect to $u$ and get $\frac{1}{6}{u}^6$.

Step:5

Simplify the expression.

Step:5.1

Expand $\frac{2}{3}(\frac{1}{6}{u}^6+C)$.

Step:5.2

Combine the constants to get $\frac{1}{9}{u}^6+C$.

Step:6

Substitute $u$ back in terms of $x$ to obtain $\frac{1}{9}(6x+1{)}^6+C$.

Knowledge Notes:

1. u-Substitution: A technique used in integration to simplify the integral by substituting a part of the integrand with a new variable $u$. This often simplifies the integral into a more recognizable form.

2. Sum Rule: A derivative rule that states the derivative of a sum is the sum of the derivatives.

3. Constant Multiple Rule: A derivative rule that allows us to take constants out of the derivative. If $c$ is a constant and $f(x)$ is a function, then $\frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]$.

4. Power Rule: A basic rule of differentiation that states if $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$.

5. Constant Rule: A rule in differentiation that states the derivative of a constant is zero.

6. Integration: The process of finding the integral of a function, which is the reverse operation of differentiation. It can be thought of as finding the area under the curve of the function.

7. Power Rule for Integration: A rule for integration that states $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C$, where $n\ne -1$ and $C$ is the constant of integration.