1.Evaluate: (1/(x*sqrt(x^2-4))
I know the answer is -1/2*arctan(2/sqrt(x^2-4)), but I am having trouble getting to this answer on my own.
I know the formula to solve it is 1/a*arctan(x/a) and that a=2, but that's all I know.
2.Find the limit of x^((ln 2)/(1+ln(x))) as x approaches negative infinity.
To evaluate the expression (1/(x*sqrt(x^2-4))), we can start by simplifying the expression.
First, let's simplify the denominator. Recall that (a^2 - b^2) can be factored as (a - b)(a + b).
So, (x^2 - 4) can be factored as (x - 2)(x + 2).
Now, let's rewrite our expression as:
1 / (x * sqrt((x - 2)(x + 2)))
To simplify this further, we can express sqrt((x - 2)(x + 2)) as sqrt(x - 2) * sqrt(x + 2).
Our expression now becomes:
1 / (x * sqrt(x - 2) * sqrt(x + 2))
Next, let's split the fraction by dividing the numerator and denominator separately:
1 / x * 1 / (sqrt(x - 2) * sqrt(x + 2))
Now, let's simplify each part separately:
1 / x can be rewritten as x^(-1).
And sqrt(x - 2) can be expressed as (x - 2)^(1/2).
Thus, our expression becomes:
x^(-1) / ((x - 2)^(1/2) * sqrt(x + 2))
Now, we can combine the terms under the square root, since (x - 2)^(1/2) * sqrt(x + 2) is the same as sqrt((x - 2)(x + 2)), which we encountered before.
Therefore, our expression simplifies to:
x^(-1) / sqrt((x - 2)(x + 2))
Now, we can apply our knowledge of the arctangent formula.
The formula states that if we have an expression in the form of 1/a * arctan(x/a), we can rewrite it as arctan(x/a)/a.
In our case, a = 2. So, we want to write our expression in the form of arctan(x/2)/2.
To do this, we can rewrite our expression as:
arctan((x-2)/2) - arctan((x+2)/2) / 2
Thus, our final expression is:
arctan((x-2)/2) - arctan((x+2)/2) / 2
This matches with the answer you provided: -1/2 * arctan(2/sqrt(x^2-4)).
Now let's move on to the second question.
To find the limit of x^((ln 2)/(1+ln(x))) as x approaches negative infinity, we can first simplify the exponent.
Using the logarithm identities, we can rewrite the exponent as:
(ln 2) / (1 + ln(x)) = (ln 2) / (ln(e) + ln(x))
Since ln(e) is equal to 1, we can simplify further:
(ln 2) / (ln(e) + ln(x)) = (ln 2) / ln(ex)
Now, we can rewrite the exponent as:
ln(2) / ln(ex)
Recall that ln(x) / ln(a) can be simplified to log base a of x.
Therefore, we have:
ln(2) / ln(ex) = log base e of 2
Since e is a positive number, we can conclude that the limit of x^((ln 2)/(1+ln(x))) as x approaches negative infinity is:
log base e of 2, which is approximately 0.693.