Title: CR_Int_11
Course: Demonstrating STACK (v4.3.7)

Linearity in integration

Integrate \[[value]\] with respect to [value].
Feedback: To solve this problem we will have to use the linearity rules for integration calc_int_linearity_rule as follows: \[\begin{array}{rcl}\displaystyle \int \left([value]\right) d[value] &\displaystyle =&\displaystyle [value]\int [value] d[value] \\ &\displaystyle =&\displaystyle [value][[value]] \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate \[[value]\] with respect to [value].
Feedback: To solve this problem we will have to use the linearity rules for integration calc_int_linearity_rule as follows: \[\begin{array}{rcl}\displaystyle \int \left([value]\right) d[value] &\displaystyle =&\displaystyle \int [value] d[value] - \int [value] d[value] \\ &\displaystyle =&\displaystyle [value] - [value] + c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate \[[value]\] with respect to [value].
Feedback: To solve this problem we will have to use the linearity rules for integration calc_int_linearity_rule as follows: \[\begin{array}{rcl}\displaystyle \int \left([value]\right) d[value] &\displaystyle =&\displaystyle \int [value] d[value] + \int [value] d[value] \\ &\displaystyle =&\displaystyle [value]\int [value] d[value] - [value]\int [value] d[value] \\ &\displaystyle =&\displaystyle [value][value] - [value][value] + c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate [value] with respect to [value].
Feedback: To solve this problem we will have to use the linearity rules for integration calc_int_linearity_rule as follows: \[\begin{array}{rcl}\displaystyle \int [[value]] d[value] &\displaystyle =&\displaystyle \int [value] d[value] + \int [value] d[value] \\ &\displaystyle =&\displaystyle [value]\int [value] d[value] + \int [value] d[value] \\ &\displaystyle =&\displaystyle [value]\left[[value]\right] + \left[[value]\right] + c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate [value] with respect to [value].
Feedback: To solve this problem we will have to use the linearity rules for integration calc_int_linearity_rule as follows: \[\begin{array}{rcl}\displaystyle \int [[value]] d[value] &\displaystyle =&\displaystyle \int [value] d[value] + \int [value] d[value] \\ &\displaystyle =&\displaystyle [value]\int [value] d[value] + [value]\int [value] d[value] \\ &\displaystyle =&\displaystyle [value]\left[[value]\right] + [value]\left[[value]\right] + c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Integrate \[[value]\] with respect to [value].
Feedback: To solve this problem we will have to use the linearity rules for integration calc_int_linearity_rule as follows: \[\begin{array}{rcl}\displaystyle \int [[value]] d[value] &\displaystyle =&\displaystyle \int [value] d[value] + \int [value] d[value] \\ &\displaystyle =&\displaystyle [value]\int [value] d[value] + \int [value] d[value] \\ &\displaystyle =&\displaystyle [value]\left[[value]\right] + \left[[value]\right] + c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Simplify and integrate [value] with respect to [value].
Feedback: Using the distributive law and the properties of exponentials alg_indices we get that \[\begin{array}{rcl}\displaystyle [value] &\displaystyle =&\displaystyle [value][value] - [value][value] \\ &\displaystyle =&\displaystyle [value] - [value] \end{array}\] So, using linearity in integration calc_int_linearity_rule, \[\begin{array}{rcl}\displaystyle \int [value] d[value] &\displaystyle =&\displaystyle \int ([value] - [value]) d[value] \\ &\displaystyle =&\displaystyle \int [value] d[value] - \int [value] d[value] \\ &\displaystyle =&\displaystyle [value] - [value] + c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]

Simplify and integrate \[\frac[value][value]\] with respect to [value].
Feedback: Using the distributive law and the properties of exponentials alg_indices we get that \[\begin{array}{rcl}\displaystyle \frac[value][value] &\displaystyle =&\displaystyle \frac[value][value] - \frac[value][value] \\ &\displaystyle =&\displaystyle [value] \end{array}\] So, using linearity in integration calc_int_linearity_rule, \[\begin{array}{rcl}\displaystyle \int \frac[value][value] d[value] &\displaystyle =&\displaystyle \int ([value] - [value]) d[value] \\ &\displaystyle =&\displaystyle \int [value] d[value] - \int [value] d[value] \\ &\displaystyle =&\displaystyle [value] - [value] + c \\ &\displaystyle =&\displaystyle [value] + c \end{array}\]