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Profª Débora Bastos
Mais fórmulas básicas
                   vn + 1            d(v n )      n − 1 dv
5 −   ∫   v n dv =
                   n + 1
                          + k
                                      dx
                                             = nv
                                                        dx


          dv                      d(lnv)   1   dv
6 −   ∫    v
             = ln v + k
                                    dx
                                         =
                                           v
                                             ⋅
                                               dx


                    av            d(a v )
7 −   ∫   a v dv =
                   ln a
                        + k
                                   dx
                                             v
                                          = a . ln a ⋅
                                                       dv
                                                       dx

                                  d(ev )
      ∫e
           v            v                      dv
8 −            dv = e       + k             v
                                         = e ⋅
                                   dx          dx
Exemplos:                        5 −    ∫ tgxdx           6 −      ∫ cotgxdx
                                                 x
    ∫ ( 2x + 5) dx 7 − ∫ 1 − 4x2 dx                                  8 − ∫ 2x dx
                   2
1 −

2 − ∫  x 5x2 − 3 dx
                 
                  9 − ∫ 3tg4x sec24xdx
            dx
                                                                   ∫e
                                                                        x2
3 −   ∫   1 − 5x
                         10 −      ∫e
                                            3x
                                                 dx      11 −                xdx

               3                       x2
           x                      e x                              1 + 2 ln x
4 −   ∫   x4 + 2
                   dx 12 −   ∫   1 + ex
                                             2
                                                     dx 13 −
                                                               ∫       x
                                                                              dx

Fórmulas decorrentes de combinações de
outras fórmulas:
            15 − ∫ tgvdv = ln(sec v) + k
            16 − ∫ cotgvdv = ln(senv) + k
Fórmulas relacionadas a funções
 trigonométricas.
                                      d(cosv)            dv
9 −  ∫ senvdv = - cos v + k             dx
                                               = −senv ⋅
                                                         dx
10 − ∫ cosvdv = senv + k              d(senv)
                                               = cos v ⋅
                                                         dv
                                         dx              dx
11 − ∫ sec2vdv = tgv + k              d(tgv)       2
                                              = sec v ⋅
                                                         dv
                                        dx               dx
12 − ∫ cosec2vdv = − cotgv + k           d(cotgv)
                                                      = cosec2v ⋅ dv
                                              dx                  dx
13 −   ∫ cosec v.cotgv.dv   = − cosec v + k
                             d(cosecv)                              dv
                                         = −cosecv ⋅ cot gv ⋅
                                dx                                  dx

14 −   ∫ sec v.tgv.dv   = sec v + k
                                      d(secv)                       dv
                                                   = secv ⋅ tgv ⋅
                                         dx                         dx
Exemplos:
    sen( lnx)
1 − ∫ x dx
2 − ∫ cos23x ⋅ sen3x ⋅ dx

3 − ∫ (4secx ⋅ tgx − 5cosec2x)dx

4 − ∫ xcosx2dx

          xsen 1 - x2
5 −   ∫         1 - x2
                            dx

            (
          tg e −3x   ) dx
6 −   ∫     e3x

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Matematica2 8

  • 2. Mais fórmulas básicas vn + 1 d(v n ) n − 1 dv 5 − ∫ v n dv = n + 1 + k dx = nv dx dv d(lnv) 1 dv 6 − ∫ v = ln v + k dx = v ⋅ dx av d(a v ) 7 − ∫ a v dv = ln a + k dx v = a . ln a ⋅ dv dx d(ev ) ∫e v v dv 8 − dv = e + k v = e ⋅ dx dx
  • 3. Exemplos: 5 − ∫ tgxdx 6 − ∫ cotgxdx x ∫ ( 2x + 5) dx 7 − ∫ 1 − 4x2 dx 8 − ∫ 2x dx 2 1 − 2 − ∫  x 5x2 − 3 dx     9 − ∫ 3tg4x sec24xdx dx ∫e x2 3 − ∫ 1 − 5x 10 − ∫e 3x dx 11 − xdx 3 x2 x e x 1 + 2 ln x 4 − ∫ x4 + 2 dx 12 − ∫ 1 + ex 2 dx 13 − ∫ x dx Fórmulas decorrentes de combinações de outras fórmulas: 15 − ∫ tgvdv = ln(sec v) + k 16 − ∫ cotgvdv = ln(senv) + k
  • 4. Fórmulas relacionadas a funções trigonométricas. d(cosv) dv 9 − ∫ senvdv = - cos v + k dx = −senv ⋅ dx 10 − ∫ cosvdv = senv + k d(senv) = cos v ⋅ dv dx dx 11 − ∫ sec2vdv = tgv + k d(tgv) 2 = sec v ⋅ dv dx dx 12 − ∫ cosec2vdv = − cotgv + k d(cotgv) = cosec2v ⋅ dv dx dx 13 − ∫ cosec v.cotgv.dv = − cosec v + k d(cosecv) dv = −cosecv ⋅ cot gv ⋅ dx dx 14 − ∫ sec v.tgv.dv = sec v + k d(secv) dv = secv ⋅ tgv ⋅ dx dx
  • 5. Exemplos: sen( lnx) 1 − ∫ x dx 2 − ∫ cos23x ⋅ sen3x ⋅ dx 3 − ∫ (4secx ⋅ tgx − 5cosec2x)dx 4 − ∫ xcosx2dx xsen 1 - x2 5 − ∫ 1 - x2 dx ( tg e −3x ) dx 6 − ∫ e3x