Evaluate ∫ ln ⁡ 2 ln ⁡ 4 e − x 1 − e − 2 x d x

Question

Evaluate      ∫          ln      ⁡      2              ln      ⁡      4                  e              −        x                    1      −              e                  −          2          x                      d  x

Nola Rivera · Accepted Answer

Observe that                          ∫                  1                      /                    4                          1                      /                    2                                      d          u                          1          −                      u            2                              =      arcsin      ⁡      y      |              1              /            4              1              /            2        =  arcsin  ⁡      1    2    −  arcsin  ⁡      1    4  But     arcsin  ⁡  x  =  arctan  ⁡      x          1      −              x        2               , so the above result equals  arcsin  ⁡      1    2    −  arcsin  ⁡      1    4    =  arctan  ⁡      1          3        −  arctan  ⁡      1          15        =      π    6    −  arctan  ⁡      1          15      which equals     arcsin  ⁡            15        4    −      π    3      …

Yahir Tucker · Accepted Answer

Notice that :      ∫          1              /            4              1              /            2                                    d                u                    1        −                  u          2                      =  −            [              cos                  −          1                    ⁡      u      ]              1              /            4              1              /            2        =      cos          −      1        ⁡            1      4        −      cos          −      1        ⁡            1      2        =      cos          −      1        ⁡            1      4        −            π      3      But we know that :      cos          −      1        ⁡  x  =      sin          −      1        ⁡      1    −          x      2      then :      ∫          1              /            4              1              /            2                                    d                u                    1        −                  u          2                      =      sin          −      1        ⁡      1    −                  1        16              −            π      3        =      sin          −      1        ⁡                    15            4        −            π      3

Evaluate <msubsup> ∫ <mrow class="MJX-TeXAtom-ORD"> ln &#x2

Answered question

Answer & Explanation

New Questions in Calculus 1