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Integrals table

Integration — is one of the main mathematical operations. On this page, the tables contain examples of the most common integrals.

As an arbitrary integration constant, the number C, which can be determined if the value of the integral is known at some point. Each function has an infinite number of antiderivatives.

Basic formulas

1.  
dx = C
2.  
a dx = ax + C      (a = const)
3.  
xn dx = xn+1n + 1 + C      (n ≠ -1)
4.  
dxx = ln |x| + C
5.  
ax dx = axln a + C
6.  
ex dx = ex + C
7.  
sin x dx = -cos x + C
8.  
cos x dx = sin x + C
9.  
dxsin2 x = -ctg x + C
10.  
dxcos2 x = tg x + C
11.  
dxa2 - x2 = arcsin xa + C = -arccos xa + C      (x < a)
12.  
dxa2 + x2 = 1a arctg xa + C = -1a arcctg xa + C
13.  
dxx2 - a2 = 12a ln x + ax - a + C      (|x| ≠ a)
14.  
dxx2 ± a2 = ln |x + x2 ± a2|
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General rules for functions integrating

c f(x) dx = c f(x) dx
[ f(x) + g(x)] dx f(x) dx g(x) dx
[ f(x) - g(x)] dx f(x) dx g(x) dx
f(x)g(x) dx = f(x) g(x) dx ∫∫ g(x) dx df(x)

Integrals of rational functions

1.  
xn dx = xn+1n + 1 + C      (n ≠ -1)
2.  
(ax + b)n dx = (ax + b)n+1a(n + 1) + C      (n ≠ -1)
3.  
dxx = ln |x| + C
4.  
dxax + b = 1a ln |ax + b| + C
5.  
ax + bcx + d dx = acx + bc - adc2 ln |cx + d| + C
6.  
dx(x + a)(x + b) = 1a - b ln |x + bx + a| + C
7.  
dxx2 - a2 = 12a ln |x - ax + a| + C
8.  
x dx(x + a)(x + b) = 1a - b (a ln |x + a| - b ln|x + b|) + C
9.  
x dxx2 - a2 = 12 ln |x2 - a2| + C
10.  
dxx2 + a2 = 1a arctg (xa) + C
11.  
x dxx2 + a2 = 12 ln |x2 + a2| + C
12.  
dx(x2 + a2)2 = 12a2 xx2 + a2 + 12a3 arctg (xa) + C
13.  
x dx(x2 + a2)2 = -12 1x2 + a2 + C
14.  
x dx(x2 + a2)3 = -14 1(x2 + a2)2 + C
15.  
dxax2 + bx + c = 1b2 - 4ac ln2ax + b - b2 - 4ac2ax + b + b2 - 4ac + C     (b2 - 4ac > 0)
16.  
dxax2 + bx + c = 14ac - b2 arctg2ax + b4ac - b2 + C     (b2 - 4ac < 0)
17.  
x dxax2 + bx + c = 12a ln|ax2 + bx + c| - b2a dxax2 + bx + c
18.  
x dxax + b = 1a2(ax + b - b ln |ax + b|) + C
19.  
x2 dxax + b = 1a3(12(ax + b)2 -2b(ax + b) + b2 ln |ax + b|) + C
20.  
dxx(ax + b) = 1b ln ax + bx + C
21.  
dxx2(ax + b) = - 1bx + ab2 ln ax + bx + C
22.  
x dx(ax + b)2 = 1a2(ln |ax + b | + bax + b) + C
23.  
x2 dx(ax + b)2 = 1a3(ax + b - 2b ln |ax + b | - b2ax + b) + C

Integrals of transcendental functions

1.  
ex dx = ex + C
2.  
ax dx = axln a + C
3.  
dxx ln x = ln |ln x| + C
4.  
xn ln x dx = xn + 1(ln xn + 1 - 1(n + 1)2) + C
5.  
eax ln x dx = eax ln xa - 1aeaxx dx
6.  
xn lnm x dx = xn + 1n + 1 lnm x - mn + 1xn lnm - 1 x dx
7.  
xnlnm x dx = -xn + 1(m - 1) lnm - 1 x + n + 1m - 1xnlnm - 1 x dx
8.  
ln x dx = x ln x - x + C
9.  
arcsin x dx = x arcsin x + 1 - x2 + C
10.  
arctg x dx = x arctg x - ln 1 + x2 + C
11.  
eax dx = eaxa + C
12.  
x eax dx = eaxa2(ax + 1) + C
13.  
axxn dx = ax(n - 1)xn - 1 + ln an - 1axxn - 1
14.  
sh(x) dx = ch(x) + C
15.  
ch(x) dx = sh(x) + C

Integrals of irrational functions

1.  
dxax + b = 2aax + b + C
2.  
ax + b dx = 23a(ax + b)1.5 + C
3.  
x dxax + b = 2(ax - 2b)3a2ax + b + C
4.  
xax + b dx = 2(3ax - 2b)15a2(ax + b)1.5 + C
5.  
dx(x + c)ax + b = 1b - ac lnax + b - b - acax + b + b - ac + C     (b - ac > 0)
6.  
dx(x + c)ax + b = 1ac - b arctgax + bac - b + C     (b - ac < 0)
7.  
ax + bcx + d dx = 1c (ax + b)(cx + d) - ad - bccac arctg a(cx + d)c(ax + b) + C
8.  
dxxax + b = 1b lnax + b - bax + b + b + C     (b > 0)
9.  
dxxax + b = 1-b arctgax + b-b + C     (b < 0)
10.  
dxx2ax + b = -ax + bbx - a2b dxxax + b
11.  
ax + bx dx = 2ax + b + b dxxax + b
12.  
a - xb + x dx = (a - x)(b + x) + (a + b)arcsinx + ba - x + C
13.  
a + xb - x dx = -(a + x)(b - x) - (a + b)arcsinb - xa + x + C
14.  
dxax2 + bx + c = 1a ln|2ax + b + a(ax2 + bx + c)| + C
15.  
dxax2 + bx + c = -1a arcsin2ax + bb2 - 4ac + C
16.  
ax2 + bx + c dx = 2ax + b4aax2 + bx + c + 4ac - b28a dxax2 + bx + c
17.  
x2 + a2 dx = x2x2 + a2 + a22 ln |x + x2 + a2| + C
18.  
x2 - a2 dx = x2x2 - a2 - a22 ln |x + x2 - a2| + C
19.  
dxx2 + a2 = ln|x + x2 + a2)| + C
20.  
dxx2 - a2 = ln|x + x2 - a2)| + C
21.  
x dxx2 + a2 = x2 + a2 + C
22.  
x2 - a2x dx = x2 - a2 + a arcsin (xa) + C
23.  
a2 - x2 dx = x2a2 - x2 + a22 arcsin (xa) + C
24.  
a2 - x2x dx = a2 - x2 + a ln (xa + a2 - x2) + C
25.  
dxa2 - x2 = arcsin (xa) + C
26.  
x dxa2 - x2 = -a2 - x2 + C
27.  
dxxa2 - x2 = 1a ln |xa + a2 - x2| + C

Integrals of trigonometric functions

1.  
sin (x) dx = -cos (x) + C
2.  
cos (x) dx = sin (x) + C
3.  
sin2 (x) dx = x2 - 14 sin (2x) + C
4.  
cos2 (x) dx = x2 + 14 sin (2x) + C
5.  
sinn (x) dx = -1n sinn - 1 (x) cos (x) + n - 1n sinn - 2 (x) dx
6.  
cosn (x) dx = 1n cosn - 1 (x) sin (x) + n - 1n cosn - 2 (x) dx
7.  
dxsin (x) = ln|tg(x2)| + C
8.  
dxcos (x) = ln|ctg(x2)| + C
9.  
dxsin2 (x) = -ctg (x) + C
10.  
dxcos2 (x) = tg (x) + C
11.  
sin (x) cos (x) dx = -14cos (2x) + C
12.  
sin2 (x) cos (x) dx = 13sin3 (x) + C
13.  
sin (x) cos2 (x) dx = -13cos3 (x) + C
14.  
sin2 (x) cos2 (x) dx = -18x - 132sin (4x) + C
15.  
tg (x) dx = -ln |cos (x)| + C
16.  
ctg (x) dx = ln |sin (x)| + C
17.  
sin (x)cos2 (x) dx = 1cos (x) + C
18.  
cos (x)sin2 (x) dx = -1sin (x) + C
19.  
sin2 (x)cos2 (x) dx = tg (x) - x + C
20.  
cos2 (x)sin2 (x) dx = -ctg (x) - x + C
21.  
sin2 (x)cos (x) dx = ln|ctg(x2)| - sin (x) + C
22.  
cos2 (x)sin (x) dx = ln|tg(x2)| + cos (x) + C
23.  
dxsin (x) cos (x) = ln|tg(x)| + C
24.  
dxsin2 (x) cos (x) = -1sin (x) + ln|ctg(x2)| + C
25.  
dxsin (x) cos2 (x) = 1cos (x) + ln|tg(x2)| + C
26.  
dxsin2 (x) cos2 (x) = tg(x) - ctg(x) + C
27.  
dxsinn (x) = -1n - 1cos (x)sinn - 1 (x) + n - 2n - 1 dxsinn - 2 (x)
28.  
tgn (x) dx = tgn - 1 (x)n - 1 tgn - 2 (x) dx
29.  
ctgn (x) dx = -ctgn - 1 (x)n - 1 ctgn - 2 (x) dx
30.  
sin (x) cosn (x) dx = -cosn + 1 (x)n + 1 + C
31.  
cos (x) sinn (x) dx = sinn + 1 (x)n + 1 + C

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