Show that integrate.[e^(-4x)*(cos 3x)] from 0-infinity =4/25
I got the answer(without applying limits) as, {(e^(-4x) )[3sin(3x) - 4cos(3x)]}/25
But when applying the upper limit what is the value of,
{(e^(-infinity) )[3sin(3*infinity) -4cos(3*infinity) ] }/25 ?
And while having a look at the answer I noticed that if the part we applied the upper limit is equal to 0,the rest,when we apply the lower limit,will give us the answer they've given..
Do e^(infinity) or sin(infinity) or cos(infinity) equal to zero by definition?
e^-∞ = 0
sin∞ and cos∞ are undefined, but lie between -1 and 1
multiplying by 0 leaves the result = zero
Thank you!
When evaluating limits involving exponential functions, sine, or cosine, it's important to note that these functions are not defined for infinite values. Division by zero is undefined, and the behaviors of these functions become unbounded as the input approaches infinity. Therefore, it is incorrect to directly substitute infinity into the expressions.
Instead, to evaluate the given integral from 0 to infinity, we need to find the limit as the upper bound approaches infinity. Let's consider each term separately and apply the limits correctly:
1. (e^(-4x)): As x approaches infinity, the exponential term e^(-4x) tends to 0. This is because the exponential function decays rapidly as the exponent becomes more negative.
2. (3sin(3x) - 4cos(3x)): Both the sine and cosine functions are bound between -1 and 1. Therefore, as the input approaches infinity, these functions oscillate between -1 and 1 but do not approach any specific value. So, we cannot apply a limit directly to these functions.
Now, let's address your observation. In the expression you provided,
{(e^(-4x))[3sin(3x) - 4cos(3x)]}/25,
as x approaches infinity, the term e^(-4x) approaches 0. If we multiply this term by (3sin(3x) - 4cos(3x)), which does not have a defined limit, the entire expression would evaluate to zero.
However, this does not mean that the integral from 0 to infinity is zero. It means that the contribution of that particular term to the integral is zero. To evaluate the integral correctly, we need to use appropriate methods such as integration by parts or a change of variables.
In this case, to show that the integral is equal to 4/25, you would need to use integration techniques rather than directly substituting values like infinity into the expression.