Consider the function f(x) = (4x + 4)/(2x^2+4x) and the function g(x)=(2x+2)/(x^2+2x) Clearly, f(x) = g(x) for any value of x and thus f and g should be thought of as merely different notation for the same function. Now find the integral of f(x)dx and the integral of g(x)dx (without factoring out 2 in the case of f). Explain why it is not a problem that you get different solutions.
INT f(x)dx= log(x)+log(x+2) + constant1
INT g(x)dx= log(x) + log(x+2) + constant2
To find the integral of f(x) and g(x), we will need to use the method of integration.
Let's start with finding the integral of f(x). We can rewrite f(x) as:
f(x) = (4x + 4)/(2x^2 + 4x)
To integrate f(x), we can split it into two separate integrals:
∫(4x + 4)/(2x^2 + 4x) dx
= ∫(4x)/(2x^2 + 4x) dx + ∫(4)/(2x^2 + 4x) dx
Now, let's look at the first integral:
∫(4x)/(2x^2 + 4x) dx
To solve this integral, we can factor out 4x from the numerator and denominator:
∫(4x)/(2x(x + 2)) dx
Next, we can cancel out the common factors of 2 and x:
∫2/(x + 2) dx
Now, we can find the integral of this simplified expression:
= 2ln|x + 2| + C1
where C1 is the constant of integration.
Now, let's move on to the second integral:
∫(4)/(2x^2 + 4x) dx
We notice that the denominator can be factored out:
∫(4)/(2x(x + 2)) dx
Similarly, we can cancel out the common factors of 2 and x:
∫2/(x(x + 2)) dx
Now, we can find the integral of this simplified expression:
= ln|x(x + 2)| + C2
where C2 is the constant of integration.
So, the integral of f(x)dx is:
∫f(x)dx = 2ln|x + 2| + ln|x(x + 2)| + C
Now, let's find the integral of g(x). We can start by rewriting g(x) as:
g(x) = (2x + 2)/(x^2 + 2x)
To integrate g(x), we can perform partial fraction decomposition:
g(x) = (2x + 2)/(x(x + 2))
Using partial fraction decomposition, we can express g(x) as:
g(x) = A/x + B/(x + 2)
Now, we can find the values of A and B. By multiplying both sides of the equation by the common denominator (x(x + 2)), we get:
2x + 2 = A(x + 2) + B(x)
Expanding the right side and equating the coefficients, we can solve for A and B:
2x + 2 = Ax + 2A + Bx
Equating the coefficients for the x terms, we get:
A + B = 2
Equating the constant terms, we get:
2A = 2
Solving these equations, we find A = 1 and B = 1.
Now, we can rewrite g(x) as the sum of two separate fractions:
g(x) = 1/x + 1/(x + 2)
Next, we can integrate each fraction separately:
∫1/x dx + ∫1/(x + 2) dx
The integral of 1/x is ln|x|:
∫1/x dx = ln|x|
The integral of 1/(x + 2) is ln|x + 2|:
∫1/(x + 2) dx = ln|x + 2|
So, the integral of g(x)dx is:
∫g(x)dx = ln|x| + ln|x + 2| + C
Now, let's address why it is not a problem that we get different solutions for the integrals of f(x) and g(x).
Even though f(x) = g(x) for any value of x, their integrals can still yield different results. This is because the integral is not just affected by the function values, but also by the form of the function.
In this case, to find the integral of f(x), we used a different approach than for g(x). Although the functions are equal, the way we dealt with them during the integration process led to different solutions. However, both solutions are valid and correct within their respective contexts.
It is important to note that when proving that two functions are equal for all values of x, we are comparing their function values and behavior, not their integrals. The integrals may differ due to differences in how the functions are written or approached during integration.