Integrate Using u-Substitution integral of 4(6x+1)^5 with respect to x
The question asks for the application of the integration technique known as u-substitution to solve the integral of an algebraic expression. Specifically, you are to integrate the function 4(6x+1)^5 with respect to x, where 4 is a constant multiplier and (6x+1)^5 represents a composite function raised to the fifth power. U-substitution involves identifying a part of the integrand (the function being integrated) that can be substituted with a new variable (commonly denoted as 'u') which simplifies the integral, followed by finding the derivative of this substitution (du) to replace the dx term. After the substitution, the integral is easier to evaluate, and once this is done, the variable u is replaced back with the original expression in terms of x to complete the integration process.
$\int 4 \left(\left(\right. 6 x + 1 \left.\right)\right)^{5} d x$
Answer
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 $\frac{d}{dx}(x^n) = nx^{n-1}$ 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:
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.
Sum Rule: A derivative rule that states the derivative of a sum is the sum of the derivatives.
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 $\frac{d}{dx}[cf(x)] = c\frac{d}{dx}[f(x)]$.
Power Rule: A basic rule of differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Constant Rule: A rule in differentiation that states the derivative of a constant is zero.
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.
Power Rule for Integration: A rule for integration that states $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$, where $n \neq -1$ and $C$ is the constant of integration.
