Is ( x 2 + 1 ) / ( x 2 − 5 x + 6 ) divisible?I'm learning single variable calculus right now and at current about integration with partial fraction. I'm stuck in a problem from few hours given in my book. The question is to integrate x 2 + 1 x 2 − 5 x + 6 .I know it is improper rational function and to make it proper rational fraction we have to divide x 2 + 1 x 2 − 5 x + 6 I'm trying from sometime but couldn't find the right solution.Please help! Thank you in advance.

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Is   (      x    2    +  1  )      /    (      x    2    −  5  x  +  6  ) divisible?I&#039;m learning single variable calculus right now and at current about integration with partial fraction. I&#039;m stuck in a problem from few hours given in my book. The question is to integrate                    x        2            +      1                      x        2            −      5      x      +      6        .I know it is improper rational function and to make it proper rational fraction we have to divide                    x        2            +      1                      x        2            −      5      x      +      6      I&#039;m trying from sometime but couldn&#039;t find the right solution.Please help! Thank you in advance.

Kaylyn Ewing · Accepted Answer

You can use long division. Another way is as follows:                                                        x              2                        +            1                                              x              2                        −            5            x            +            6                                              =                              (                          x              2                        −            5            x            +            6            )            +            (            5            x            −            5            )                                              x              2                        −            5            x            +            6                                                            =                                            x              2                        −            5            x            +            6                                              x              2                        −            5            x            +            6                          +                              5            x            −            5                                              x              2                        −            5            x            +            6                                                            =        1        +                              5            x            −            5                                              x              2                        −            5            x            +            6                              As next step notice that      x    2    −  5  x  +  6  =  (  x  −  2  )  (  x  −  3  )and expand in partial fractions:                                          5            x            −            5                                              x              2                        −            5            x            +            6                                              =                  A                      x            −            2                          +                  B                      x            −            3                                            Where         A         and         B         are constants

osmane5e · Accepted Answer

∫                            x          2                +        1                              x          2                −        5        x        +        6              d    x    =    ∫                  (                  x          2                −        5        x        +        6        )        +        (        5        x        −        5        )                              x          2                −        5        x        +        6              d    x    =    ∫    1    d    x    +    5    ∫                  (        x        −        1        )                    (        x        −        2        )        (        x        −        3        )              d    x  Now Using Partial fraction Method::                    x        −        1                    (        x        −        2        )        ⋅        (        x        −        3        )              =          A              (        x        −        2        )              +          B              (        x        −        3        )            So we get   x  −  1  =  A  (  x  −  3  )  +  B  (  x  −  2  )  =  (  A  +  B  )  x  +  (  −  3  A  −  2  B  )After Camparing the coefficient, we get       A    +    B    =    1   and   3  A  +  2  B  =  1So we get       A    =    −    1   and       B    =    2  So we get                               x          2                +        1                              x          2                −        5        x        +        6              d    x    =    x    +    5    ∫          [      −              1                  x          −          2                    +              2                  x          −          3                    ]        d    x

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