00:01
Hello everyone, for the given functions we need to calculate the convolution.
00:05
The function f of t is given as cos t and g of t is given as minus 3 minus 3t and f asterisk of g of t can be written as cos t multiplied by minus 3 minus 3t.
00:31
So, here in the first part of the equation we need to replace t with tau in the function f and replace t with t minus tau in the second function and we need to multiply them together.
00:47
So, we have f of tau as cos tau and g of t minus tau to be minus 3 minus 3 of t minus tau.
01:04
So, simplifying this we have this value to be minus 3 minus 3t plus 3 tau.
01:12
So now we can multiply these two functions.
01:17
So it would be f of tau multiplied by g of tau is cos tau multiplied by minus 3 minus 3t plus 3 tau.
01:35
So this is the answer for the first part of the equation.
01:39
Now let us move on to the next part of the equation where we need to find the definite integral for the given multiplication of these two functions with respect to tau.
01:52
So now i to be equal to integral of cos of tau multiplied by minus 3 minus 3t plus 3 tau multiplied by d tau.
02:12
So now taking this function as f and this function as g and by integration by parts.
02:19
So now applying this that is integral of f dot g can be written as f into integral of g minus integral of f dash multiplied by integral of...