Find the integral: 25 x 2 − 4 x d x

Question

Find the integral:            25              x                  2                    −      4        x    d  x

hanfydded1c · Accepted Answer

Use Trigonometric SubstitutionLet   x  =      2    5    sec  ⁡  u  ,  d  x  =            2      sec      ⁡      u      tan      ⁡      u        5    d  uSubstitute variables from above.  ∫            25      (              2        5            sec      ⁡      u              )                  2                    −      4                      2        5            sec      ⁡      u        ×            2      sec      ⁡      u      tan      ⁡      u        5    d  uSimplify.  ∫  2      tan          2        ⁡  u  d  uUse Constant Factor Rule:   ∫  c  f  (  x  )  d  x  =  c  ∫  f  (  x  )  d  x  .  2  ∫      tan          2        ⁡  u  d  uUse Pythagorean Identities:       tan          2        ⁡  x  =      sec          2        ⁡  x  −  1  2  ∫      sec          2        ⁡  u  −  1  d  uUse Sum Rule:   ∫  f  (  x  )  +  g  (  x  )  d  x  =  ∫  f  (  x  )  d  x  +  ∫  g  (  x  )  d  x  .  2  (  ∫      sec          2        ⁡  u  d  u  +  ∫  −  1  d  u  )The derivative of   tan  ⁡  u is       sec          2        ⁡  u  2  (  tan  ⁡  u  +  ∫  −  1  d  u  )Use this rule:   ∫  a  d  x  =  a  x  +  C  2  (  tan  ⁡  u  −  u  )From the earlier steps, we know that:  sec  ⁡  u  =      5    2    x  tan  ⁡  u  =      (          5      2        x          )              2              −    1  Substitute the above back into the original integral.  2  (      (          5      2        x          )              2              −    1    −      sec          −      1        ⁡  (      5    2    x  )  )Add constant.  2  (                    25                  x                      2                              4        −    1    −      sec          −      1        ⁡  (      5    2    x  )  )  +  C

Find the integral: int sqrt(25x^2-4)/x dx

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