Title: CR_Int_10
Course: Demonstrating STACK (v4.3.7)

Integrals of basic functions

Integrate \[[value]^[value]\] with respect to [value].
Feedback: This problem can be solved using the rule for integration \[\int x^n dx = \frac{1}{n+1}x^{n+1}+c\] where c is a constant of integration. \[\begin{array}{rcl}\displaystyle \int [value] d[value] &\displaystyle =&\displaystyle \left[\frac{1}{[value]+1}\right]x^{[value]+1}+c\\ &\displaystyle =&\displaystyle \left[\frac{1}{\left([value]\right)}\right]x^[value]+c\\ &\displaystyle =&\displaystyle [value]+c \end{array}\]

Integrate \[[value]^[value]\] with respect to [value].
Feedback: This problem can be solved using the rule for integration \[\int x^n dx = \frac{1}{n+1}x^{n+1}+c\] where c is a constant of integration. \[\begin{array}{rcl}\displaystyle \int [value]^[value] d[value] &\displaystyle =&\displaystyle \left[\frac{1}{[value]+1}\right]x^{[value]+1}+c\\ &\displaystyle =&\displaystyle \left[\frac{1}{\left([value]\right)}\right]x^[value]+c\\ &\displaystyle =&\displaystyle [value]+c \end{array}\]

Integrate [value] with respect to [value].
Feedback: Note that the integral of \(\cos(x)\) is \(\sin(x)\) So, \[\begin{array}{rcl}\displaystyle \int [value] d[value] &\displaystyle =&\displaystyle \frac{1}[value][value]+c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate [value] with respect to [value].
Feedback: Note that the integral of \(\sin(x)\) is \(-\cos(x)\) So, \[\begin{array}{rcl}\displaystyle \int [value] d[value] &\displaystyle =&\displaystyle \frac{1}[value][[value]]+c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate [value] with respect to [value].
Feedback: Note that the integral of \(sin(x)\) is \(-sin(x)\) So, \[\begin{array}{rcl}\displaystyle \int [value] d[value] &\displaystyle =&\displaystyle \frac{1}[value][[value]]+c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate \([value]\) with respect to [value].
Feedback: Note that the integral of \(\cos(x)\) is \(\sin(x)\) So, \[\begin{array}{rcl}\displaystyle \int \cos([value]) d[value] &\displaystyle =&\displaystyle \frac{1}[value]\sin([value])+c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate [value] with respect to [value].
Feedback: The integral of [value] with respect to [value] is a standard result, but it does not fit in the general pattern \[ \int x^n \mathrm{d}x = \frac{x^{n+1}}{n+1}+c.\] Instead \[\int \frac{1}{x} \mathrm{d}x = \int x^{-1} \mathrm{d}x = \ln(|x|)+c.\] This is interesting, but the reasons for this are normally discussed in a real analysis course.

Integrate [value] with respect to [value].
Feedback: This problem can be solved by inspection since the derivative of [value] with respect to [value] is [value]. So, \[\begin{array}{rcl}\displaystyle \int [value] d[value] &\displaystyle =&\displaystyle \frac{1}[value][value]+c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]