The Antiderivative of: (2x+3)/((x^3)+3x)
This one is devinately integration by partial fractions but im not sure what i am doing wrong. PLEASE HELP.
To find the antiderivative of the given function, you are correct that you need to use the method of partial fractions. This method involves decomposing the given rational function into simpler fractions and then integrating each fraction separately.
Here are the steps to follow:
1. Factorize the denominator: The denominator of the given function is x^3 + 3x. You can factor out an x from each term: x(x^2 + 3). So, the denominator becomes x(x^2 + 3).
2. Write the decomposition: The numerator (2x + 3) cannot be factored further. So, we can write it as two fractions with unknown constants as numerators: A/x + (Bx + C)/(x^2 + 3).
3. Cross multiplication and equate coefficients: Multiply both sides of the equation by the common denominator, x(x^2 + 3), to eliminate the denominators. Then, equate the coefficients of the corresponding powers of x.
(2x + 3) = A(x^2 + 3) + (Bx + C)x.
Simplify the equation: 2x + 3 = Ax^2 + 3A + Bx^2 + Cx.
Now, equate the coefficients:
2x + 3 = (A + B)x^2 + Cx + 3A.
Equating coefficients of x^2: A + B = 0.
Equating coefficients of x: C = 2.
Equating constant terms: 3A = 3.
From A + B = 0, we get B = -A.
From C = 2, we get C = 2.
And from 3A = 3, we get A = 1.
4. Rewrite the decomposition: Now that we have values for A, B, and C, we can rewrite our decomposition as follows:
(2x + 3)/(x(x^2 + 3)) = 1/x + (-x + 2)/(x^2 + 3).
5. Integrate each term separately: Now, integrate each term separately.
∫(1/x) dx = ln|x| + C1, where C1 is the constant of integration.
∫(-x + 2)/(x^2 + 3) dx = -0.5ln|x^2 + 3| + C2, where C2 is the constant of integration.
6. Combine the results: Add the two integrated terms to obtain the antiderivative of the given function.
∫(2x + 3)/((x^3) + 3x) dx = ln|x| - 0.5ln|x^2 + 3| + C.
So, the antiderivative of (2x + 3)/((x^3) + 3x) is ln|x| - 0.5ln|x^2 + 3| + C, where C is the constant of integration.