Solving integral using trig substitution tan ⁡ ( x / 2 ) = tI have problems with solving the following integral: ∫ sin ⁡ x − cos ⁡ x sin ⁡ x + cos ⁡ x d xCould anybody please help me to find the solution

Question

Solving integral using trig substitution   tan  ⁡  (  x      /    2  )  =  tI have problems with solving the following integral:  ∫                    sin        ⁡        x        −        cos        ⁡        x                    sin        ⁡        x        +        cos        ⁡        x                d  xCould anybody please help me to find the solution

Nollothwnfcm · Accepted Answer

∫            sin      ⁡      (      x      )      −      cos      ⁡      (      x      )              sin      ⁡      (      x      )      +      cos      ⁡      (      x      )           d  x  =Substitute   u  =  sin  ⁡  (  x  )  +  cos  ⁡  (  x  ) and   d  u  =  (  cos  ⁡  (  x  )  −  sin  ⁡  (  x  )  )     d  x:  −  ∫      1    u       d  u  =  −  ln  ⁡      |    u    |    +  C  =  −  ln  ⁡      |    sin    ⁡    (    x    )    +    cos    ⁡    (    x    )    |    +  C

Regina Ewing · Accepted Answer

Notice,  ∫            sin      ⁡      x      −      cos      ⁡      x              sin      ⁡      x      +      cos      ⁡      x           d  x  =  ∫            −      (      cos      ⁡      x      −      sin      ⁡      x      )              sin      ⁡      x      +      cos      ⁡      x        d  x  =  −  ∫            d      (      sin      ⁡      x      +      cos      ⁡      x      )              sin      ⁡      x      +      cos      ⁡      x        =      −    ln    ⁡          |        sin    ⁡    x    +    cos    ⁡    x          |        +    C

Solving integral using trig substitution tan ⁡ ( x <mrow class="MJX-TeXA

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