SOLUTION: {{{int(sqrt(3 + t^2),dt,sin(x),2)}}}, calculate dy/dx. I know how to do the integral itself, I just don't know how to find dy/dx. Please help me solve this.

Algebra ->  Equations -> SOLUTION: {{{int(sqrt(3 + t^2),dt,sin(x),2)}}}, calculate dy/dx. I know how to do the integral itself, I just don't know how to find dy/dx. Please help me solve this.      Log On


   

Question 1001098: int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29, calculate dy/dx. I know how to do the integral itself, I just don't know how to find dy/dx. Please help me solve this.
Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


Since one of the limits of integration is a function of x, the evaluated integral is a function of x. Compute the integral and then take the derivative of the resulting function.

John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29,   calculate dy/dx.
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Let F(x) = int%28sqrt%283+%2B+t%5E2%29%2Cdt%2Csin%28x%29%2C2%29.

Since the lower limit of integration is a function of  x,  the derivative of the integral is taken
with the sign  "minus"  derivative of the function representing the lower limit,  multiplied by the value of the function
under the integral symbol taken at the current value of  x:

F'(x) = -%28d%28sin%28x%29%29%29%2F%28dx%29 . sqrt%283+%2B+x%5E2%29 = -cos%28x%29 . sqrt%283+%2B+x%5E2%29.


For more clarity,  if

F(x) = int%28f%28t%29%2Cdt%2Cg%28x%29%2Cb%29,

where  b = const  and the lower limit of integration  g(x)  is a function of  x,  then

F'(x) = -g'(x)*f(x).


For the reference,  see  this Wikipedia article.