Evaluate the following definite integral:
integral at a = -1, b=2
-4dx/(9x^2+30x+25)
Would I have to separate them in 3 terms as:
-4 ∫1/9x^2 + ∫1/30x + ∫1/25
resulting in: -4/(3x^3)+ (15x^2)+ C?
and from there I can replace a and b
f(a) - f(b)?
Thank you
Booo!
1/(a+b+c) is NOT 1/a + 1/b + 1/c
What you need to do is let
u = 3x+5 and you have
du = 3 dx, so dx = du/3
x in [-1,2] means u in [2,11]
giving you
∫[2,11] -4/u^2 du/3
-4/3 ∫[2,11] u^-2 du
That should be ever so simple.
THanks Steve, but why 3x+5??
because you have 9x^2 + 30x + 25 = (3x+5)^2
To evaluate the definite integral ∫[a, b] (-4dx)/(9x^2+30x+25), you can follow these steps:
1. Begin by noticing that the integrand is a rational function. One approach is to first use partial fraction decomposition to express the integrand in the form A/(x-p) + B/(x-q), where p and q are the roots of the denominator polynomial.
2. The denominator, 9x^2+30x+25, can be factored as (3x+5)(3x+5) or (3x+5)^2. Therefore, the partial fraction decomposition is of the form A/(3x+5) + B/(3x+5).
3. To determine the values of A and B, you can use the method of equating coefficients. Multiply the entire equation by (3x+5)^2 as follows:
-4 = A(3x+5) + B(3x+5)
4. Now, simplify the equation above and gather like terms:
-4 = (3A+3B)x + (5A+5B)
5. Equate the coefficients of like powers of x on both sides of the equation:
For the constant term: -4 = 5A + 5B
For the x term: 0 = 3A + 3B
6. Solve the equations above simultaneously to find that A=2 and B=-2.
7. Now that you have the partial fraction decomposition, you can rewrite the integral as follows:
∫[-1, 2] (-4dx)/(9x^2+30x+25) = ∫[-1, 2] (2/(3x+5) - 2/(3x+5)) dx
8. Distribute the integral to both terms:
= ∫[-1, 2] 2/(3x+5) dx - ∫[-1, 2] 2/(3x+5) dx
9. Since the integrals are identical, you can write it as:
= 2∫[-1, 2] 2/(3x+5) dx
10. Evaluate the integral:
= 2 [ln|3x+5|] [-1, 2]
11. Substitute the upper and lower limits into the antiderivative:
= 2 [ln|3(2)+5| - ln|3(-1)+5|]
12. Simplify the expression:
= 2 [ln|11| - ln|2|]
13. Further simplify:
= 2 ln(11/2)
14. Finally, calculate the value:
= 2 ln(5.5)
So, the value of the definite integral ∫[-1, 2] (-4dx)/(9x^2+30x+25) is 2 ln(5.5).