Find the integral: ∫ ( 3 x − 2 ) 2 x 3 d x

Question

Find the integral:  ∫            (      3      x      −      2              )                  2                            x              3              d  x

Rylan White · Accepted Answer

Use   (  a  −  b      )    2    =      a    2    −  2  a  b  +      b    2   to expand the expression in the numerator:  ∫            9              x                  2                    −      12      x      +      4              x              3              d  x We have three terms in the numerator, so let’s separate the fraction into three fractions:  ∫            9              x                  2                            x              3              −            12      x              x              3              +      4          x              3              d  xNotice that there’s   x in the numerator and denominator of the first two fractions, so we can simplify those:   ∫      9    x    −      12          x              2              +      4          x              3              d  xUse the property of integral   ∫  f  (  x  )  ±  g  (  x  )  d  x  =  ∫  f  (  x  )  d  x  ±  ∫  g  (  x  )  d  x   ∫      9    x    d  x  −  ∫      12          x              2              d  x  +  ∫      4          x              3              d  xTime to use the integration rules! We’ll use   ∫      a    x    d  x  =  a  ×  ln  ⁡      (          |        x          |        )   to evaluate the integral:  9  ln  ⁡      (          |        x          |        )    −  ∫      12          x              2              d  x  +  ∫      4          x              3              d  xTo evaluate the second integral, first use   ∫  a  ×  f  (  x  )  d  x  =  a  ×  ∫  f  (  x  )  d  x  9  ln  ⁡      (          |        x          |        )    −  12  ×  ∫      1          x              2              d  x  +  ∫      4          x              3              d  xNow, use   ∫      1          x      n        d  x  =  −      1          (      n      −      1      )      ×              x                  n          −          1                     to evaluate the integral:  9  ln  ⁡      (          |        x          |        )    −  12  ×  (  −      1    x    )  +  4  ×  (  −      1          2              x        2              )   Simplify the expressions:  9  ln  ⁡      (          |        x          |        )    +      12    x    −      2          x      2      Since the derivative of a constant is zero, we need to add the constant of integration   C to include all possible anti-derivative functions in the result:  9  ln  ⁡      (          |        x          |        )    +      12    x    −      2          x      2        +  C  ,  C  ∈      R  So, the indefinite integral   ∫            (      3      x              –            2              )                  2                            x              3              d  x is equal to:  9  ln  ⁡      (          |        x          |        )    +      12    x    −      2          x      2        +  C  ,  C  ∈      R

Find the integral: int ((3x - 2)^{2))/(x^{3))dx

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