Неопределенные интегралы. Желтухин В.С. - 5 стр.

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1)
Z
0 · dx = C . 2)
Z
1 · dx =
Z
dx = x + C.
3)
Z
x
α
dx =
x
α+1
α + 1
+ C, α 6= −1.
4)
Z
1
x
dx =
Z
dx
x
= ln |x| + C, x 6= 0.
5)
Z
dx
1 + x
2
= arctg x + C.
6)
Z
dx
p
1 − x
2
= arcsin x + C, |x| < 1.
7)
Z
dx
√
x
2
− 1
= ln(x +
p
x
2
− 1) + C, |x| > 1.
8)
Z
dx
√
x
2
+ 1
= ln(x +
p
x
2
+ 1) + C.
9)
Z
a
x
dx =
a
x
ln a
+ C, a > 0, a 6= 1. 10)
Z
e
x
dx = e
x
+ C.
11)
Z
sin x dx = −cos x + C. 12)
Z
cos x dx = sin x + C.
13)
Z
dx
sin
2
x
= −ctg x + C. 14)
Z
dx
cos
2
x
= tg x + C.
15)
Z
sh x dx = ch x + C. 16)
Z
ch x dx = sh x + C.
17)
Z
1
sh
2
x
dx = −cth x + C. 18)
Z
1
ch
2
x
dx = th x + C.
                                                                Òàáëèöà 1

         Îñíîâíûå èíòåãðàëû îò ýëåìåíòàðíûõ ôóíêöèé
     Z                                       Z           Z
1)    0 · dx = C.                            2) 1 · dx =       dx = x + C.
              Z
                           xα+1
           3) xα dx =            + C, α 6= −1.
                          α+1
              Z          Z
                 1           dx
           4)       dx =         = ln |x| + C, x 6= 0.
                 x            x
              Z
                    dx
           5)            = arctg x + C.
                 1 + x2
              Z
                     dx
           6) p             = arcsin x + C, |x| < 1.
                    1 − x2
              Z                       p
                     dx
           7) √            = ln(x + x2 − 1) + C, |x| > 1.
                    x2 − 1
              Z                       p
                     dx
           8) √            = ln(x + x2 + 1) + C.
                     2
                    x +1
  Z               x                              Z
                 a
9) ax dx =           + C, a > 0, a 6= 1. 10) ex dx = ex + C.
                ln a
    Z                                            Z
11) sin x dx = − cos x + C.                  12) cos x dx = sin x + C.
    Z                                            Z
          dx                                          dx
13)        2   = − ctg x + C.                14)           = tg x + C.
       sin x                                        cos2 x
    Z                                            Z
15) sh x dx = ch x + C.                      16) ch x dx = sh x + C.
    Z                                             Z
          1                                            1
17)           dx  =  − cth x +  C.            18)          dx = th x + C.
       sh2 x                                         ch2 x




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