Evaluate the integral
((6x+7)/(x^(2)-8x+25))dx. Please and thank you.
The integral sign can be input in HTML as
& i n t ;
(remove the embedded spaces).
See response to other post:
http://www.jiskha.com/display.cgi?id=1322572957
Go on:
wolframalpha dot com
When page be open in rectangle type:
integrate (6x+7)/(x^2-8x+25)
and click option =
After few seconds when you see result click option: Show steps
On wolframalpha dot com you can practice any kind of calculus.
To evaluate the integral of ((6x+7)/(x^2 - 8x + 25))dx, we can use a method known as partial fractions. The first step is to factorize the denominator.
Let's start by factoring x^2 - 8x + 25. The quadratic equation can be written as (x - 4)^2 + 9.
Now, let's write the given expression as a sum of two fractions:
((6x+7)/(x^2 - 8x + 25)) = A/(x - 4) + B/(x - 4)^2 + 9
To determine the values of A and B, we need to find a common denominator. In this case, the common denominator is (x - 4)^2 + 9.
Now, we can rewrite the equation as:
(6x + 7) = A((x - 4)^2 + 9) + B(x - 4)
Next, we multiply both sides of the equation by the common denominator:
(6x + 7) = A((x - 4)^2 + 9) + B(x - 4)((x - 4)^2 + 9)
Expanding and simplifying the right side gives:
6x + 7 = A(x^2 - 8x + 25 + 9) + B(x^3 - 8x^2 + 25x - 9x - 100)
Simplifying further:
6x + 7 = A(x^2 - 8x + 34) + B(x^2 - 8x + 25)
Now, we can equate the coefficients of the like terms on both sides of the equation.
For the term with x^2 coefficient: 0 = A + B
For the term with x coefficient: 6 = -8A - 8B
For the constant term: 7 = 34A + 25B
Solving this system of equations will give us the values of A and B. Once we have the values for A and B, we can rewrite the integral as:
∫(6x + 7)/(x^2 - 8x + 25) dx = ∫A/(x - 4) dx + ∫B/(x - 4)^2 dx
Using the properties of integrals, we can solve these two integrals separately.
∫A/(x - 4) dx can be evaluated to A ln| x - 4 | + C, where C is the constant of integration.
∫B/(x - 4)^2 dx can be evaluated to -B/(x - 4) + C, where C is again the constant of integration.
Finally, we can substitute the values of A and B into the respective integrals and add the two results to obtain the final answer.