I still do not understand how to solve the following problem. Please show step-by-step.
x = Integral (0 to v) dv/(q^2-v^2) where q = constant
the answer is --> x = 1/2q ln (q+v/q-v)
1) Where 1/2q came from?
2) Where ln came from? Is there a formula for it?
Please help!!!!!!!
I explained this yesterday. You use the method of partial fractions, to rewrite
1/(q^2-v^2) as the sum of terms with
(q+v)and (q-v) in the denominator. When you integrate those two separate terms, you get the difference of two log terms.
Look up the method in your text or verify the integal in a table of integrals.
[1/(2q)] is a factor appears when the method of partial fractions is correctly applied.
"ln" is term that designates the natural (base e) logarithm of whatever follows. ln x is the integral of dx/x
ln (q+v) is the integral of dv/(q+v), with q being a constant. When you do the integration, you get the ln(q+v) - ln (q-v). Using the rules of logarithms, this can be written ln[(q+v)/(q-v)]. The 1/2q factor comes from the step of converting 1/(q^2-v^2) to two partial fractions.
To solve the given integral, you can use the method of partial fractions. Here's a step-by-step explanation:
1) Start by factoring the denominator:
q^2 - v^2 = (q + v)(q - v)
2) Rewrite the integral using the factored form of the denominator:
x = ∫ dv / [(q + v)(q - v)]
3) Now, apply partial fractions to break down the integrand:
Since the denominator can be factored as (q + v)(q - v), we can write:
1 / [(q + v)(q - v)] = A / (q + v) + B / (q - v)
where A and B are constants to be determined.
4) Multiply both sides of the equation by the common denominator (q + v)(q - v):
1 = A(q - v) + B(q + v)
Expanding this equation, we get:
1 = (A + B)q + (-A + B)v [equation 1]
5) To find the values of A and B, equate the coefficients of q and v on both sides:
From equation 1, we have:
A + B = 0 [equation 2]
-A + B = 1 [equation 3]
6) Solve equations 2 and 3 simultaneously. Adding equations 2 and 3, we get:
2B = 1
Therefore, B = 1/2.
Substituting B = 1/2 into equation 2, we find:
A + (1/2) = 0
Hence, A = -1/2.
7) Now that we have found the values of A and B, we can rewrite the original integral using partial fractions:
x = ∫ [(-1/2) / (q + v)] + [(1/2) / (q - v)] dv
8) Integrate each term separately:
∫ (-1/2) / (q + v) dv = (-1/2) ln(q + v)
∫ (1/2) / (q - v) dv = (1/2) ln(q - v)
9) Combining the results and factoring out the constants, we obtain:
x = (-1/2) ln(q + v) + (1/2) ln(q - v)
10) Simplify the expression by combining the logarithmic terms:
x = ln[(q - v)/(q + v)] / 2
11) To further simplify, note that ln[(q - v)/(q + v)] can be expressed as ln(q - v) - ln(q + v).
Therefore, we have:
x = [ln(q - v) - ln(q + v)] / 2
12) Finally, rearrange the terms and factor out the constant 1/2:
x = 1/2 [ln(q + v) - ln(q - v)]
And that is the correct answer to the integral:
x = 1/2 ln[(q + v)/(q - v)]
To answer your specific questions:
1) The factor of 1/2 in the final answer comes from the integration of the constant coefficient 1/2.
2) The "ln" in the final answer represents the natural logarithm. It is a standard mathematical function used to calculate the inverse of the exponential function. The natural logarithm does not have a simple formula like some other functions, but it is widely used in mathematics and has specific properties that allow us to solve various problems involving exponential and logarithmic functions.