Title: CR_Int_12
Course: Demonstrating STACK (v4.3.7)

Evaluating definite integrals

Evaluate the following definite integral:
\[ \int^[value]_[value][value]d[value]. \]
Feedback: If \[\int f(x) dx = F(x)+c\] then \[\int^a_b f(x) dx = F(a) - F(b).\] So, in order to calculate the definite integral of [value] with respect to [value] we first calculate its integral: \[\int [value] d[value] = [value]+c.\] So, let \[F([value]) = [value]\] Then, \[\begin{array}{rcl}\displaystyle \int^[value]_[value] [value] d[value] &\displaystyle =&\displaystyle F([value]) - F([value]) \\ &\displaystyle =&\displaystyle [value] - [value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Evaluate the following definite integral:
\[ \int^[value]_[value][value]d[value]. \]
Feedback: If \[\int f(x) dx = F(x)\] then \[\int^a_b f(x) dx = F(a) - F(b).\] So, in order to calculate the definite integral of [value] with respect to [value] we first calculate its integral (Note: we don't need to consider a constant of integration): \[\int [value] d[value] = [value].\] So, let \[F([value]) = [value]\] Then, \[\begin{array}{rcl}\displaystyle \int^[value]_[value] [value] d[value] &\displaystyle =&\displaystyle F([value]) - F([value]) \\ &\displaystyle =&\displaystyle [value] - [value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Evaluate the following definite integral: \[ \int^[value]_[value][value]d[value] + \int^[value]_[value][value]d[value]. \]
Feedback: If \[\int f(x) dx = F(x)\] then \[\int^a_b f(x) dx = F(a) - F(b).\] So, in order to calculate the definite integral of [value] with respect to [value] we first calculate its integral (Note: we don't need to consider a constant of integration): \[\int [value] d[value] = [value].\] So, let \[F([value]) = [value]\] Then, \[\begin{array}{rcl}\displaystyle \int^[value]_[value] [value] d[value] &\displaystyle =&\displaystyle F([value]) - F([value]) \\ &\displaystyle =&\displaystyle [value] - [value] \\ &\displaystyle =&\displaystyle [value] \end{array}\] and \[\begin{array}{rcl}\displaystyle \int^[value]_[value] [value] d[value] &\displaystyle =&\displaystyle F([value]) - F([value]) \\ &\displaystyle =&\displaystyle [value]-[value] \\ &\displaystyle =&\displaystyle [value] \end{array}\] So, \[\begin{array}{rcl}\displaystyle \int^[value]_[value] [value] d[value] + \int^[value]_[value] [value] d[value] &\displaystyle =&\displaystyle [value]+ [value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]

Evaluate the following definite integral:
\[ \int^[value]_[value][value]d[value].\]
Feedback: If \[\int f(x) dx = F(x)\] then \[\int^a_b f(x) dx = F(a) - F(b).\] So, in order to calculate the definite integral of [value] with respect to [value] we first calculate its integral (Note: we don't need to consider a constant of integration): \[\int [value] d[value] = [value].\] So, let \[F([value]) = [value]\] Then, \[\begin{array}{rcl}\displaystyle \int^[value]_[value] [value] d[value] &\displaystyle =&\displaystyle F([value]) - F([value]) \\ &\displaystyle =&\displaystyle [value]-[value] \\ &\displaystyle =&\displaystyle [value] \end{array}\]