$\int(2x+6)^5\,dx$

Full solution

\int(2x+6)^5\,dx

Recognize the integral: Recognize the integral to be solved. $\newline$ We need to find the integral of $(2x+6)^5$ with respect to $x$ .
Apply power rule: Apply the power rule for integration. $\newline$ The power rule for integration states that the integral of $x^n$ with respect to $x$ is $(x^{(n+1)})/(n+1) + C$ , where $C$ is the constant of integration. However, since we have a linear term $(2x+6)$ instead of just $x$ , we need to adjust the rule accordingly.
Use substitution: Use substitution to simplify the integral. $\newline$ Let $u = 2x + 6$ , then $du = 2dx$ . This means that $dx = \frac{du}{2}$ .
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ . The integral becomes $\frac{1}{2} \times \int u^5 \, du$ , since we have to divide by $2$ to account for the substitution of $dx$ .
Apply power rule for u: Apply the power rule for integration to the integral in terms of $u$ . Using the power rule, we get $(\frac{1}{2}) \times (\frac{u^6}{6}) + C$ .
Substitute back $x$ : Substitute back the original variable $x$ into the integral. $\newline$ Replace $u$ with $2x + 6$ to get $(1/2) \times ((2x + 6)^{6}/6) + C$ .
Simplify expression: Simplify the expression. $\newline$ We can simplify the expression to $(\frac{1}{12}) \times (2x + 6)^6 + C$ .