Can someone show me a step by step process, along with explanations on how to do this problem?
Thank you. :)
Evaluate the integral ∫ [(x^2 - 1)/(x^2 + 1)]dx?
x - 2tan^-1(x) + c
Thank you. :)
I was wondering, did you break up the fractions and then integrate it?
that's correct. A simple division shows that
(x^2-1)/(x^2+1) = 1 - 2/(x^2+1)
Certainly! To evaluate the given integral, we can use a method called partial fraction decomposition. This method involves breaking down the given rational function into simpler fractions that can be integrated more easily.
Step 1: Express the rational function as the sum of partial fractions.
In this case, the denominator of the given rational function, (x^2 + 1), is a quadratic expression that cannot be factored further using real numbers. Therefore, we need to use complex numbers.
Let's assume there are two unknown constants, A and B, such that we can write the rational function as:
(x^2 - 1)/(x^2 + 1) = A(x + i) + B(x - i)
Step 2: Simplify and find the values of A and B.
To determine the values of A and B, we need to find a common denominator and then equate the numerators:
(x^2 - 1) = A(x + i)(x - i) + B(x - i)(x + i)
Expanding and simplifying the right side of the equation:
x^2 - 1 = A(x^2 - i^2) + B(x^2 - (-i)^2)
x^2 - 1 = A(x^2 + 1) + B(x^2 + 1)
Now, equate the coefficients of like terms on both sides:
1) x^2 terms: 1 = A + B
2) Constant terms: -1 = A + B
Solving this system of equations, we find A = -1/2 and B = 1/2.
Therefore, we can rewrite the given rational function as:
(x^2 - 1)/(x^2 + 1) = (-1/2)(x + i) + (1/2)(x - i)
Step 3: Evaluate the integral.
Now that we have expressed the rational function as partial fractions, we can integrate each term separately.
∫ [(x^2 - 1)/(x^2 + 1)]dx = ∫ [(x + i)/2] - ∫ [(x - i)/2]
Integrating each term separately:
= (1/2)∫(x + i)dx - (1/2)∫(x - i)dx
= (1/2)[(1/2)x^2 + ix] - (1/2)[(1/2)x^2 - ix]
= (1/4)x^2/2 + (1/4)ix - (1/4)x^2/2 + (1/4)ix
= (1/4)x^2 + (1/4)ix - (1/4)x^2 + (1/4)ix
= (1/2)ix
So, the solution to the integral is:
∫ [(x^2 - 1)/(x^2 + 1)]dx = (1/2)ix + C
Note: C represents the constant of integration.