- #1
tahayassen
- 270
- 1
[tex]1.\int { x } dx=x\int { 1 } dx\\ 2.\int { t } dx=t\int { 1 } dx\\ 3.\int _{ x }^{ x+1 }{ x } dt=x\int _{ x }^{ x+1 }{ 1 } dt[/tex]
Which of the equations are correct?
Which of the equations are correct?
tahayassen said:[tex]1.\int { x } dx=x\int { 1 } dx\\ 2.\int { t } dx=t\int { 1 } dx\\ 3.\int _{ x }^{ x+1 }{ x } dt=x\int _{ x }^{ x+1 }{ 1 } dt[/tex]
Which of the equations are correct?
Mark44 said:And 1 is incorrect. The following is a property of integrals:
##\int k~f(x)~dx = k\int f(x)~dx##, for k a constant, but there is no property that says you can move a variable across the integral sign.
No, x is the variable of integration so we cannot take it outside the integral.tahayassen said:[tex]1.\int { x } dx=x\int { 1 } dx[/tex]
If we know that t is independent of x, then both integrals are "tx+ C". If t is a function of x then the first is still "tx+ C" but the other depends upon exactly what function of x t is.[tex] 2.\int { t } dx=t\int { 1 } dx[/tex]
If x is independent of the variable of integration, t, both of those are the same and are equal to x(x+1- x)= x. If x is a function of t, then the left depends upon exactly what function of t x is while the right is still x.[tex] 3.\int _{ x }^{ x+1 }{ x } dt=x\int _{ x }^{ x+1 }{ 1 } dt[/tex]
Which of the equations are correct?
No, not all variables can be moved outside of the integrand. Only constants and independent variables can be moved outside of the integrand. Variables that are dependent on the integration variable cannot be moved.
Moving variables outside of the integrand can help simplify the integration process and make it easier to solve the integral. It can also help identify patterns and make the integration more manageable.
Yes, there are certain rules and restrictions for moving variables outside of the integrand. For example, when moving a variable outside of the integrand, its power must be divided by the new location of the variable. Additionally, the order of the variables must be maintained.
Yes, variables can be moved outside of the integrand in all types of integration, including definite and indefinite integrals. However, the process and rules may differ slightly depending on the type of integration.
Moving variables outside of the integrand can change the form and complexity of the integral, making it easier to solve. It can also help identify and simplify the integral, leading to a more accurate and efficient result.