Evaluate integral(5x-5/3x^2-8x-3) dx.
Use partial fractions
(5x-5)/(3x^2 - 8x - 3)
= (5x-5)/((x-3)(3x+1))
let (5x-5)/((x-3)(3x+1))
= A/(x-3) + B/(x-3)
= (A(x-3) + B(3x+1))/((x-3)(3x+1))
so
5x - 5 = A(x-3) + B(3x+1)
let x=3 ----> 10 = B(10) ---> B=1
let x = -1/3
-20/3 = A(-10/3) ----> A = 2
so
(5x-5)/((3x^2 - 8x - 3)
= 2/(x-3) + 1/(3x+1)
and the integral of that is
2ln(x-3) + (1/3)ln(3x+1) + c
To evaluate the integral ∫(5x - 5) / (3x^2 - 8x - 3) dx, we can start by factoring the denominator.
The denominator factors as (3x + 1)(x - 3).
So, the integral becomes ∫(5x - 5) / (3x + 1)(x - 3) dx.
Next, we need to decompose the fraction into partial fractions.
We write the expression as A/(3x + 1) + B/(x - 3), where A and B are constants to be determined.
To find A and B, we can perform the following algebraic manipulation:
(5x - 5) = A*(x - 3) + B*(3x + 1)
Expanding the right side of the equation:
5x - 5 = (A + 3B)x + (-3A + B)
Equating the coefficients of x on both sides:
5 = A + 3B
Equating the constant terms on both sides:
-5 = -3A + B
Solving these two equations simultaneously, we find A = 8/5 and B = -7/5.
Now that we have the partial fraction decomposition, the integral can be written as:
∫(8/5)/(3x + 1) - (7/5)/(x - 3) dx
To further simplify, we can break down this integral into two separate integrals:
∫(8/5)/(3x + 1) dx - ∫(7/5)/(x - 3) dx
The first integral, ∫(8/5)/(3x + 1) dx, can be solved by making a substitution:
Let u = 3x + 1. Then, du/dx = 3, and dx = du/3.
Substituting these values into the integral:
∫(8/5)/(3x + 1) dx = ∫(8/5)/u * (du/3) = (8/5)*(1/3) * ∫du/u
= (8/15)ln|u| + C1
= (8/15)ln|3x + 1| + C1.
For the second integral, ∫(7/5)/(x - 3) dx, we can solve it using another substitution:
Let v = x - 3. Then, dv/dx = 1, and dx = dv.
Substituting these values into the integral:
∫(7/5)/(x - 3) dx = ∫(7/5)/v * dv = (7/5)ln|v| + C2
= (7/5)ln|x - 3| + C2.
Finally, putting everything together, the integral becomes:
∫(5x - 5) / (3x^2 - 8x - 3) dx = (8/15)ln|3x + 1| - (7/5)ln|x - 3| + C,
where C = C1 + C2 is the constant of integration.
To evaluate the integral of the given function, we can start by factoring the denominator of the fraction:
3x^2 - 8x - 3
To factorize the quadratic expression, we need to find two numbers whose product is equal to the product of the coefficient of x^2 term (3) and the constant term (-3). In this case, the product is -9.
The numbers that satisfy this condition are -9 and 1. Next, we rewrite the middle term (-8x) as the sum of these two numbers:
-8x = -9x + x
Now, we can factor by grouping:
3x^2 - 9x + x - 3
Factoring out the greatest common factor from the first two terms and the last two terms:
3x(x - 3) + 1(x - 3)
Now, we can see that (x - 3) is a common factor:
(x - 3)(3x + 1)
Now that we have factored the denominator as (x - 3)(3x + 1), we can rewrite the integral as:
∫(5x - 5)/(x - 3)(3x + 1) dx
To evaluate this integral, we need to use partial fractions decomposition. This involves expressing the fraction as a sum of simpler fractions. In this case, we can write:
(5x - 5)/((x - 3)(3x + 1)) = A/(x - 3) + B/(3x + 1)
To determine the values of A and B, we can find a common denominator and equate the numerators:
(5x - 5) = A(3x + 1) + B(x - 3)
Now, let's solve for A and B. We can do this by expanding the right side and comparing the coefficients of x:
5x - 5 = (3A + B)x + (A - 3B)
Equating the coefficients of x:
5 = 3A + B
-5 = A - 3B
We now have a system of equations. Solving this system, we find A = -2 and B = 1.
Substituting these values back into the partial fractions decomposition, we have:
(5x - 5)/((x - 3)(3x + 1)) = -2/(x - 3) + 1/(3x + 1)
Now, we can integrate each part separately:
∫(-2/(x - 3)) dx + ∫(1/(3x + 1)) dx
Integrating -2/(x - 3) with respect to x:
-2 ∫(1/(x - 3)) dx
To integrate this, we can use the natural logarithm function:
-2 ln|x - 3| + C1
Next, we integrate 1/(3x + 1) with respect to x:
(1/3) ∫(1/(x + 1/3)) dx
Again, we can use the natural logarithm function:
(1/3) ln|3x + 1| + C2
Finally, combining the two results, we have:
∫(5x - 5)/((x - 3)(3x + 1)) dx = -2 ln|x - 3| + (1/3) ln|3x + 1| + C
where C = C1 + C2 is the constant of integration.