arl146 said:Homework Statement
determine the indefinite integral: ∫ (4x+10)/(9x^2+24x+16) dx
Homework Equations
partial fractions technique
The Attempt at a Solution
i know it's partial fractions and i thought i did it right but i got the wrong answer.
(4x+10)/(9x^2+24x+16) = (4x+10)/(3x+4)^2 = A/(3x+4) + B/(3x+4)^2
is that part right before I continue?
As others have told you that is correct. Now multiply both sides by (3x+4)^2 to get (4x+10)(3x+ 4)^2= A(3x+ 4)+ B. Taking x= -4/3 gives an easy solution for B. Take x to be any other number, say, x= 0, to get an equation for A.arl146 said:Homework Statement
determine the indefinite integral: ∫ (4x+10)/(9x^2+24x+16) dx
Homework Equations
partial fractions technique
The Attempt at a Solution
i know it's partial fractions and i thought i did it right but i got the wrong answer.
(4x+10)/(9x^2+24x+16) = (4x+10)/(3x+4)^2 = A/(3x+4) + B/(3x+4)^2
is that part right before I continue?
The process for integrating a fraction with a quadratic denominator, such as (4x+10)/(9x^2+24x+16), involves breaking it down into partial fractions. This means expressing the fraction as a sum of simpler fractions with linear denominators.
The constants in the partial fractions are determined by equating the coefficients of like terms on both sides of the equation. This involves solving a system of linear equations to find the values of the constants.
Yes, any rational function (a fraction with polynomials in the numerator and denominator) can be expressed as partial fractions. However, it may not always be necessary or beneficial to do so.
The purpose of using partial fractions is to simplify the integration process. It allows us to break down a complex fraction into simpler ones that can be integrated more easily. It is also helpful in solving differential equations and evaluating indefinite integrals.
Partial fractions cannot be used for integration if the denominator of the fraction cannot be factored into linear terms. In such cases, other integration techniques, such as substitution or integration by parts, may be necessary.