How do you find the indefinite integral of #int (5x^2 – 4x + 7)/((x^2+1)(2x – 1)) dx#?

Question

Samantha Adkison · Accepted Answer

#I=-2arctanx+5/2ln|2x-1| +C# #I=int (5x^2 – 4x + 7)/((x^2+1)(2x – 1)) dx# #(Ax+B)/(x^2+1)+C/(2x – 1)=((Ax+B)(2x-1)+C(x^2+1))/((x^2+1)(2x – 1)) =# #=(2Ax^2+2Bx-Ax-B+Cx^2+C)/((x^2+1)(2x – 1)) =# #=(x^2(2A+C)+x(2B-A)+(-B+C))/((x^2+1)(2x – 1)) = T# If we compare integrand with expression #T#, we can write: #2A+C=5# #2B-A=-4 => A=2B+4# #-B+C=7 => C=B+7# #2(2B+4)+B+7=5# #5B=-10 => B=-2# #A=0# #C=5# #I=int (-2/(x^2+1)+5/(2x-1))dx# #I=-2arctanx+5/2ln|2x-1| +C#

Aurora Alvarado · Answer

undefined To find the indefinite integral of $\int \frac{5x^2 - 4x + 7}{(x^2 + 1)(2x - 1)} dx$, you can use partial fraction decomposition followed by standard integration techniques. After decomposing the rational function into partial fractions, you can integrate each term separately.

Here are the steps:

1. Decompose the rational function $\frac{5x^2 - 4x + 7}{(x^2 + 1)(2x - 1)}$ into partial fractions.
2. Once decomposed, integrate each term separately.
3. Combine the integrals of the partial fractions to find the indefinite integral of the original function.

The decomposition and integration steps might involve algebraic manipulation and substitution techniques. Once integrated, you should have the indefinite integral of the given function.

Emilia Aguilar · Answer

undefined To find the indefinite integral of $\int \frac{{5x^2 - 4x + 7}}{{(x^2+1)(2x - 1)}} \, dx$, you can use partial fraction decomposition to split the fraction into simpler fractions. After that, you can integrate each term separately. Once integrated, you can combine the results to find the overall indefinite integral.