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).