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.
To find the integrals of f(x)dx and g(x)dx, we first need to simplify the expressions and then apply the integral rules.
Let's start with f(x) = (4x + 4)/(2x^2 + 4x). To integrate this function, we can follow these steps:
Step 1: Simplify the expression.
f(x) = (4x + 4)/(2x^2 + 4x)
= 4(x + 1)/(2x(x + 2))
= 2(x + 1)/(x(x + 2))
Step 2: Split the fraction into partial fractions.
Since the denominator has irreducible quadratic factors (x and x + 2), we can write the expression as follows:
f(x) = A/x + B/(x + 2), where A and B are constants to be determined.
Multiplying each side by the common denominator (x(x + 2)), we get:
2(x + 1) = A(x + 2) + Bx
Expanding and simplifying this equation, we obtain:
2x + 2 = Ax + 2A + Bx
Comparing the coefficients of x on both sides, we have:
2 = A + B
And comparing the constant terms, we get:
2 = 2A
Solving these equations, we find A = 1 and B = 1.
Therefore, we can write f(x) as:
f(x) = 1/x + 1/(x + 2)
Step 3: Integrate the partial fractional expressions.
∫(1/x)dx = ln|x| + C1 (where C1 is the constant of integration)
∫(1/(x + 2))dx = ln|x + 2| + C2 (where C2 is the constant of integration)
Therefore, the integral of f(x)dx is:
∫f(x)dx = ln|x| + ln|x + 2| + C
Now, let's move on to g(x) = (2x + 2)/(x^2 + 2x). Following the same steps, we can simplify and integrate g(x)dx:
Step 1: Simplify the expression.
g(x) = (2x + 2)/(x^2 + 2x)
= 2(x + 1)/(x(x + 2))
Step 2: Split the fraction into partial fractions.
g(x) = A/x + B/(x + 2), where A and B are constants to be determined.
We solve for A and B by multiplying each side by the common denominator:
2(x + 1) = A(x + 2) + Bx
Expanding and simplifying, we get:
2x + 2 = (A + B)x + 2A
Comparing coefficients, we have:
2 = A + B
2 = 2A
Solving these equations yields A = 1 and B = 1.
Therefore, we can write g(x) as:
g(x) = 1/x + 1/(x + 2)
Step 3: Integrate the partial fractional expressions.
∫(1/x)dx = ln|x| + C1 (where C1 is the constant of integration)
∫(1/(x + 2))dx = ln|x + 2| + C2 (where C2 is the constant of integration)
Therefore, the integral of g(x)dx is:
∫g(x)dx = ln|x| + ln|x + 2| + C
Now, let's address the fact that we obtained the same solutions for the integrals of f(x) and g(x), even though the original functions may look different.
It is important to note that the integrals of two functions can still be the same, despite the functions appearing different, as long as they have the same antiderivative. In mathematics, antiderivatives are not unique - there can be multiple functions with the same derivative.
In this case, even though f(x) may seem different from g(x), they are, in fact, equivalent functions because f(x) = g(x) for any value of x. Thus, their antiderivatives (integrals) will also be the same.
In conclusion, the fact that we obtained the same solutions for the integrals of f(x) and g(x) is not a problem because they are different notations for the same function.