Find the integral: ∫ sec 3 ⁡ x d x

Question

Find the integral:  ∫      sec          3        ⁡  x  d  x

Leo Robinson · Accepted Answer

Use Integration by Parts on   ∫      sec          3        ⁡  x  d  xLet   u  =  sec  ⁡  x  ,  d  v  =      sec          2        ⁡  x  ,  d  u  =  sec  ⁡  x  tan  ⁡  x  d  x  ,  v  =  tan  ⁡  xSubstitute the above into   u  v  −  ∫  v  d  u  .  sec  ⁡  x  tan  ⁡  x  −  ∫      tan          2        ⁡  x  sec  ⁡  x  d  xUse Pythagorean Identities:       tan          2        ⁡  x  =      sec          2        ⁡  x  −  1  sec  ⁡  x  tan  ⁡  x  −  ∫  (      sec          2        ⁡  x  −  1  )  sec  ⁡  x  d  xExpand.  sec  ⁡  x  tan  ⁡  x  −  ∫      sec          3        ⁡  x  −  sec  ⁡  x  d  xUse Sum Rule:  ∫  f  (  x  )  +  g  (  x  )  d  x  =  ∫  f  (  x  )  d  x  +  ∫  g  (  x  )  d  x  .  sec  ⁡  x  tan  ⁡  x  −  ∫      sec          3        ⁡  d  x  +  ∫  sec  ⁡  x  d  xSet it as equal to the original integral   ∫      sec          3        ⁡  x  d  x  ∫      sec          3        ⁡  x  d  x  =  sec  ⁡  x  tan  ⁡  x  −  ∫      sec          3        ⁡  x  d  x  +  ∫  sec  ⁡  x  d  xAdd   ∫      sec          3        ⁡  x  d  x to both sides.  ∫      sec          3        ⁡  x  d  x  +  ∫      sec          3        ⁡  x  d  x  =  sec  ⁡  x  tan  ⁡  x  +  ∫  sec  ⁡  x  d  xSimplify    ∫      sec          3        ⁡  x  d  x  +  ∫      sec          3        ⁡  x  d  x to   2  ∫      sec          3        ⁡  x  d  x  2  ∫      sec          3        ⁡  x  d  x  =  sec  ⁡  x  tan  ⁡  x  +  ∫  sec  ⁡  x  d  xDivide both sides by   2  ∫      sec          3        ⁡  x  d  x  =            sec      ⁡      x      tan      ⁡      x      +      ∫      sec      ⁡      x      d      x        2  Original integral solved.            sec      ⁡      x      tan      ⁡      x      +      ∫      sec      ⁡      x      d      x        2  Use Trigonometric Integration: the integral of   sec  ⁡  x  ln  ⁡  (  sec  ⁡  x  +  tan  ⁡  x  )            sec      ⁡      x      tan      ⁡      x      +      ln      ⁡      (      sec      ⁡      x      +      tan      ⁡      x      )        2              sec      ⁡      x      tan      ⁡      x      +      ln      ⁡      (      sec      ⁡      x      +      tan      ⁡      x      )        2    +  C

Find the integral: int sec^3 xdx

Answered question

Answer & Explanation

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