FUNCTION | INDEFINITE INTEGRAL |
f (x) = 0 |
∫ f (x) dx = c |
f (x) = 1 |
∫ f (x) dx = 1 + c |
f (x) = a |
∫ f (x) dx = a + c |
f (x) = x |
∫ f (x) dx = (1/2)x2 + c |
f (x) = ax |
∫ f (x) dx = (1/2)ax2 + c |
f (x) = ax2 |
∫ f (x) dx = (1/3)ax3 + c |
f (x) = ax3 |
∫ f (x) dx = (1/4)ax4 + c |
f (x) = ax4 |
∫ f (x) dx = (1/5)ax5 + c |
f (x) = ax−1 |
∫ f (x) dx = a ln |x| + c |
f (x) = ax−2 |
∫ f (x) dx = −ax−1 + c |
f (x) = ax−3 |
∫ f (x) dx = −(1/2)ax−2 + c |
f (x) = ax−4 |
∫ f (x) dx = (−1/3)ax−3 + c |
f (x) = (ax + b)1/2 |
∫ f (x) dx = (2/3)(ax + b)3/2a−1 + c |
f (x) = (ax + b)−1/2 |
∫ f (x) dx = 2(ax + b)1/2a−1 + c |
f (x) = (ax + b)−1 |
∫ f (x) dx = a−1[ln (ax + b)] + c |
f (x) = (ax + b)−2 |
∫ f (x) dx = −a−1(ax + b)−1 + c |
f (x) = (ax + b)−3 |
∫ f (x) dx = −(1/2)a−1(ax + b)−2 + c |
f (x) = (ax + b)n where n ≠ −1 |
∫ f (x) dx = (ax + b)n+1(an + a)−1 + c |
f (x) = x(ax + b)1/2 |
∫ f (x) dx = (1/15)a−2(6ax − 4b)(ax + b)3/2 + c |
f (x) = x(ax + b)−1/2 |
∫ f (x) dx = (1/3)a−2(4ax − 4b)(ax + b)1/2 + c |
f (x) = x(ax + b)−1 |
∫ f (x) dx = a−1x − a−2b ln (ax + b) + c |
f (x) = x(ax + b)−2 |
∫ f (x) dx = b(a3x + a2b)−1 + a−2 ln (ax + b) + c |
f (x) = x2(ax + b)−1 |
∫ f (x) dx = [(ax + b)2/2a3] − a−3(2abx + 2b2) + (a−3b2) ln (ax + b) + c |
f (x) = x2(ax + b)−2 |
∫ f (x) dx = a−2x + a−3b − a−3b2(ax + b)−1 − 2a−3b ln (ax + b) + c |
f (x) = (ax2 + bx)−1 |
∫ f (x) dx = (b−2a) ln [(ax + b) / x)] − (bx)−1 + c |
f (x) = (ax3 + bx2)−1 |
∫ f (x) dx = {ln [x(ax + b)−1]}/b + c |
f (x) = (ax + b)(rx + s) |
∫ f (x) dx = (br − as)−1 ln [(ax + b)−1(rx + s)] + c |
f (x) = (ax + b)(rx + s)−1 |
∫ f (x) dx = ar−1x + r−2(br − as) ln (rx + s) + c |
f (x) = (x2 + a2)1/2 |
∫ f (x) dx = (x/2)(x2 + a2)1/2 + (1/2)a2 ln [x + (x2 + a2)1/2] + c |
f (x) = (x2 − a2)1/2 |
∫ f (x) dx = (x/2)(x2 − a2)1/2 + (1/2)a2 ln [x + (x2 − a2)1/2] + c |
f (x) = (a2 − x2)1/2 |
∫ f (x) dx = (x/2)(a2 − x2)1/2 + (1/2)a2 sin−1 (a−1x) + c |
f (x) = (x2 + a2)−1/2 |
∫ f (x) dx = ln [x + (x2 + a2)1/2] + c |
f (x) = (x2 − a2)−1/2 |
∫ f (x) dx = ln [x + (x2 − a2)1/2] + c |
f (x) = (a2 − x2)−1/2 |
∫ f (x) dx = sin−1 (a−1x) + c |
f (x) = (x2 + a2)−1 |
∫ f (x) dx = a−1 tan−1 (a−1x) + c |
f (x) = (x2 − a2)−1 |
∫ f (x) dx = (1/2)a−1 ln [(x + a)−1(x − a)] + c |
f (x) = (a2 − x2)−1 where |a| > |x| |
∫ f (x) dx = (1/2)a−1 ln [(a − x)−1(a + x)] + c |
f (x) = (x2 + a2)−2 |
∫ f (x) dx = (2a2x2 + 2a4)−1x + (1/2)a−3 tan−1 (a−1x) + c |
f (x) = (x2 − a2)−2 |
∫ f (x) dx = (−x)(2a2x2 − 2a4)−1 − (1/4)a−3 ln [(x + a)−1(x − a)] + c |
f (x) = (a2 − x2)−2 where |a| > |x| |
∫ f (x) dx = −x(2a4 − 2a2x2)−1 + (1/4)a−3 ln [(a − x)−1(a + x)] + c |
f (x) = x(x2 + a2)1/2 |
∫ f (x) dx = (1/3)(x2 + a2)3/2 + c |
f (x) = x(x2 − a2)1/2 |
∫ f (x) dx = (1/3)(x2 − a2)3/2 + c |
f (x) = x(a2 − x2)1/2 |
∫ f (x) dx = (−1/3)(a2 − x2)3/2 + c |
f (x) = x(x2 + a2)−1/2 |
∫ f (x) dx = (x2 + a2)1/2 + c |
f (x) = x(x2 − a2)−1/2 |
∫ f (x) dx = (x2 − a2)1/2 + c |
f (x) = x(a2 − x2)−1/2 |
∫ f (x) dx = −(a2 − x2)1/2 + c |
f (x) = x(x2 + a2)−1 |
∫ f (x) dx = (1/2) ln (x2 + a2) + c |
f (x) = x(x2 − a2)−1 |
∫ f (x) dx = (1/2) ln (x2 − a2) + c |
f (x) = x(a2 − x2)−1 where |a| > |x| |
∫ f (x) dx = −(1/2) ln (a2 − x2) + c |
f (x) = x(x2 + a2)−2 |
∫ f (x) dx = (−2x2 − 2a2)−1 + c |
f (x) = x(x2 − a2)−2 |
∫ f (x) dx = −(1/2)(x2 − a2)−1 + c |
f (x) = x(a2 − x2)−2 where |a| > |x| |
∫ f (x) dx = (1/2)(a2 − x2)−1 + c |
f (x) = x2(x2 + a2)1/2 |
∫ f (x) dx = (x/4)(x2 + a2)3/2−(1/8)a2x(x2 + a2)1/2 − (1/8)a4 ln [x + (x2 + a2)1/2] + c |
f (x) = x2(x2 − a2)1/2 |
∫ f (x) dx = (x/4)(x2 − a2)3/2 + (1/8)a2x(x2 − a2)1/2 − (1/8)a4 ln [x + (x2 − a2)1/2] + c |
f (x) = x2(a2 − x2)1/2 |
∫ f (x) dx = −(x/4)(a2 − x2)3/2 +(1/8)a2x(a2 − x2)1/2 +(1/8)a4 sin−1 (a−1x) + c |
f (x) = x2(x2 + a2)−1/2 |
∫ f (x) dx = (x/2)(x2 + a2)1/2 − (1/2)a2 ln [x + (x2 + a2)1/2] + c |
f (x) = x2(x2 − a2)−1/2 |
∫ f (x) dx = (x/2)(x2 − a2)1/2 + (1/2)a2 ln [x + (x2 − a2)1/2] + c |
f (x) = x2(a2 − x2)−1/2 |
∫ f (x) dx = −(x/2)(a2 − x2)1/2 + (1/2)a2 sin−1 (a−1x) + c |
f (x) = x2(x2 + a2)−1 |
∫ f (x) dx = x − a tan−1 (a−1x) + c |
f (x) = x2(x2 − a2)−1 |
∫ f (x) dx = x + (a/2) ln [(x + a)−1(x − a)] + c |
f (x) = x2(a2 − x2)−1 where |a| > |x| |
∫ f (x) dx = −x + (a/2) ln [(a − x)−1(a + x)] + c |
f (x) = x2(x2 + a2)−2 |
∫ f (x) dx = −x(2x2 + 2a2)−1 + (2a)−1 tan−1 (a−1x) + c |
f (x) = x2(x2 − a2)−2 |
∫ f (x) dx = −x(2x2 − 2a2)−1 + (1/4)a−1 ln [(x + a)−1(x − a)] + c |
f (x) = x2(a2 − x2)−2 where |a| > |x| |
∫ f (x) dx = x(2a2 − 2x2)−1 − (1/4)a−1 ln [(a − x)−1(a + x)] + c |
f (x) = axn |
∫ f (x) dx = axn+1(n + 1)−1 + c provided that n ≠ −1 |
f (x) = ag (x) |
∫ f (x) dx = a ∫ g (x) dx + c |
f (x) = g (x) + h (x) |
∫ f (x) dx = ∫ g (x) dx + ∫ h (x) dx + c |
f (x) = h (x)g′(x) |
∫ f (x) dx = g (x) h (x) − ∫ g′ (x) (x) + c |
f (x) = ex |
∫ f (x) dx = ex + c |
f (x) = aebx |
∫ f (x) dx = aebx / b + c |
f (x) = x−1ebx |
∫ f (x) dx = ln x + c + bx + (2! · 2)−1b2x2 + (3! · 3)−1b3x3 + (4! · 4)−1b4x4 + … |
f (x) = xebx |
∫ f (x) dx = b−1xebx + b−2ebx + c |
f (x) = x2ebx |
∫ f (x) dx = b−1x2ebx − 2b−2xebx + 2b−3ebx + c |
f (x) = ln −1 x |
∫ f (x) dx = ln (ln x) + ln x + c + (2! · 2)−1 ln2 x + (3! · 3)−1 ln3 x + … |
f (x) = x−2 ln x |
∫ f (x) dx = −x−1 ln x − x−1 + c |
f (x) = x−1 ln x |
∫ f (x) dx = (1/2) ln2 x + c |
f (x) = ln x |
∫ f (x) dx = x ln x − x + c |
f (x) = x ln x |
∫ f (x) dx = (x2/2) ln x − (1/4)x2 + c |
f (x) = x2 ln x |
∫ f (x) dx = (x3/3) ln x − (1/9)x3 + c |
f (x) = ln2 x |
∫ f (x) dx = x ln2 x − 2x ln x + 2x + c |
f (x) = sin x |
∫ f (x) dx = −cos x + c |
f (x) = cos x |
∫ f (x) dx = sin x + c |
f (x) = tan x |
∫ f (x) dx = ln |sec x| + c |
f (x) = csc x |
∫ f (x) dx = ln |tan (x/2)| + c |
f (x) = sec x |
∫ f (x) dx = ln |sec x + tan x| + c |
f (x) = cot x |
∫ f (x) dx = ln |sin x| + c |
f (x) = sin ax |
∫ f (x) dx = −a−1 cos ax + c |
f (x) = cos ax |
∫ f (x) dx = a−1 sin ax + c |
f (x) = tan ax |
∫ f (x) dx = a−1 ln (sec ax) + c |
f (x) = csc ax |
∫ f (x) dx = a−1 ln [tan (ax/2)] + c |
f (x) = sec ax |
∫ f (x) dx = a−1 ln [tan (π/4 + ax/2)] + c |
f (x) = cot ax |
∫ f (x) dx = a−1 ln (sin ax) + c |
f (x) = sin2 x |
∫ f (x) dx = (1/2){x − [(1/2) sin (2x)]} + c |
f (x) = cos2 x |
∫ f (x) dx = (1/2){x + [(1/2) sin (2x)]} + c |
f (x) = tan2 x |
∫ f (x) dx = tan x − x + c |
f (x) = csc2 x |
∫ f (x) dx = − cot x + c |
f (x) = sec2 x |
∫ f (x) dx = tan x + c |
f (x) = cot2 x |
∫ f (x) dx = − cot x − x + c |
f (x) = sin2 ax |
∫ f (x) dx = (x/2) − (1/4)a−1 (sin 2ax) + c |
f (x) = cos2 ax |
∫ f (x) dx = (x/2) + (1/4)a−1 (sin 2ax) + c |
f (x) = tan2 ax |
∫ f (x) dx = a−1 tan ax − x + c |
f (x) = csc2 ax |
∫ f (x) dx = −a−1 cot ax + c |
f (x) = sec2 ax |
∫ f (x) dx = a−1 tan ax + c |
f (x) = cot2 ax |
∫ f (x) dx = −a−1 cot ax − x + c |
f (x) = x sin ax |
∫ f (x) dx = a−2 sin ax − a−1x cos ax + c |
f (x) = x cos ax |
∫ f (x) dx = a−2 cos ax + a−1x sin ax + c |
f (x) = x2 sin ax |
∫ f (x) dx = 2a−2x sin ax+ (2a−3 − a−1x2) cos ax + c |
f (x) = x2 cos ax |
∫ f (x) dx = 2a−2x cos ax+ (a−1x2 − 2a−3) sin ax + c |
f (x) = (sin x cos x)−2 |
∫ f (x) dx = 2 cot 2x + c |
f (x) = (sin x cos x)−1 |
∫ f (x) dx = ln (tan x) + c |
f (x) = sin x cos x |
∫ f (x) dx = (1/2) sin2 x + c |
f (x) = sin2 x cos2 x |
∫ f (x) dx = (x/8) − (1/32) sin 4x + c |
f (x) = (sin ax cos ax)−2 |
∫ f (x) dx = 2a−1 cot 2ax + c |
f (x) = (sin ax cos ax)−1 |
∫ f (x) dx = a−1 ln (tan ax) + c |
f (x) = sin ax cos ax |
∫ f (x) dx = (1/2)a−1 sin2 ax + c |
f (x) = sin2 ax cos2 ax |
∫ f (x) dx = (x/8) − (1/32) (a−1) sin 4ax + c |
f (x) = sec x tan x |
∫ f (x) dx = sec x + c |
f (x) = arcsin x = sin−1 x |
∫ f (x) dx = x sin−1 x + (1 − x2)1/2 + c |
f (x) = arccos x = cos−1 x |
∫ f (x) dx = x cos−1 x − (1 − x2)1/2 + c |
f (x) = arctan x = tan−1 x |
∫ f (x) dx = x tan−1 x − (1/2) ln (1 + x2) + c |
f (x) = arccsc x = csc−1 x |
∫ f (x) dx = x csc−1 x − ln [x + (x2 − 1)1/2] + c when −π/2 < csc−1 x < 0 |
|
∫ f (x) dx = x csc−1 x + ln [x + (x2 − 1)1/2] + c when 0 < csc−1 x < π/2 |
f (x) = arcsec x = sec−1 x |
∫ f (x) dx = x sec−1 x − ln [x + (x2 − 1)1/2] + c when 0 < sec−1 x < π/2 |
|
∫ f (x) dx = x sec−1 x + ln [x + (x2 − 1)1/2] + c when π/2 < sec−1 x < π |
f (x) = arccot x = cot−1 x |
∫ f (x) dx = x cot−1 x + (1/2) ln (1 + x2) + c |
f (x) = sinh x |
∫ f (x) dx = cosh x + c |
f (x) = cosh x |
∫ f (x) dx = sinh x + c |
f (x) = tanh x |
∫ f (x) dx = ln |cosh x| + c |
f (x) = csch x |
∫ f (x) dx = ln |tanh (x/2)| + c |
f (x) = sech x |
∫ f (x) dx = 2 tan−1 ex + c |
f (x) = coth x |
∫ f (x) dx = ln |sinh x| + c |
f (x) = sinh ax |
∫ f (x) dx = a−1 cosh ax + c |
f (x) = cosh ax |
∫ f (x) dx = a−1 sinh ax + c |
f (x) = tanh ax |
∫ f (x) dx = a−1 ln |cosh ax| + c |
f (x) = csch ax |
∫ f (x) dx = a−1 ln |tanh (ax/2)| + c |
f (x) = sech ax |
∫ f (x) dx = 2a−1 tan−1 eax + c |
f (x) = coth ax |
∫ f (x) dx = a−1 ln |sinh ax| + c |
f (x) = sinh2 x |
∫ f (x) dx = (1/2) sinh x cosh x − x/2 + c |
f (x) = cosh2 x |
∫ f (x) dx = (1/2) sinh x cosh x + x/2 + c |
f (x) = tanh2 x |
∫ f (x) dx = x − tanh x + c |
f (x) = csch2 x |
∫ f (x) dx = −coth x + c |
f (x) = sech2 x |
∫ f (x) dx = tanh x + c |
f (x) = coth2 x |
∫ f (x) dx = x − coth x + c |
f (x) = sinh2 ax |
∫ f (x) dx = (1/2)a−1 sinh ax cosh ax − x/2 + c |
f (x) = cosh2 ax |
∫ f (x) dx = (1/2)a−1 sinh ax cosh ax + x/2 + c |
f (x) = tanh2 ax |
∫ f (x) dx = x − a−1 tanh ax + c |
f (x) = csch2 ax |
∫ f (x) dx = −a−1 coth ax + c |
f (x) = sech2 ax |
∫ f (x) dx = a−1 tanh ax + c |
f (x) = coth2 ax |
∫ f (x) dx = x − a−1 coth ax + c |
f (x) = (sinh x)−1 |
∫ f (x) dx = ln |tanh (x/2)| + c |
f (x) = (cosh x)−1 |
∫ f (x) dx = 2 tan−1 ex + c |
f (x) = (sinh ax)−1 |
∫ f (x) dx = a−1 ln |tanh (ax/2)| + c |
f (x) = (cosh ax)−1 |
∫ f (x) dx = 2a−1 tan−1 eax + c |
f (x) = (sinh x)−2 |
∫ f (x) dx = coth x + c |
f (x) = (cosh x)−2 |
∫ f (x) dx = tanh x + c |
f (x) = (sinh ax)−2 |
∫ f (x) dx = a−1 coth ax + c |
f (x) = (cosh ax)−2 |
∫ f (x) dx = a−1 tanh ax + c |
f (x) = (sinh x cosh x)−2 |
∫ f (x) dx = −2 coth 2x + c |
f (x) = (sinh x cosh x)−1 |
∫ f (x) dx = ln |tanh x| + c |
f (x) = sinh x cosh x |
∫ f (x) dx = (1/2) sinh2 x + c |
f (x) = sinh2 x cosh2 x |
∫ f (x) dx = (1/32) sinh 4x − (1/8)x + c |
f (x) = (sinh ax cosh ax)−2 |
∫ f (x) dx = −2a−1 coth 2ax + c |
f (x) = (sinh ax cosh ax)−1 |
∫ f (x) dx = a−1 ln |tanh ax| + c |
f (x) = sinh ax cosh ax |
∫ f (x) dx = (1/2)a−1 sinh2 ax + c |
f (x) = sinh2 ax cosh2 ax |
∫ f (x) dx = (1/32)a−1 sinh 4ax − (1/8)x + c |
f (x) = arcsinh x = sinh−1 x |
∫ f (x) dx = x sinh−1 x − (x2 + 1)1/2 + c |
f (x) = arccosh x = cosh−1 x |
∫ f (x) dx = x cosh−1 x + (x2 − 1) 1/2 + c |
|
when cosh−1 x < 0 |
|
∫ f (x) dx = x cosh−1 x − (x2 − 1)1/2 + c |
|
when cosh−1 x > 0 |
f (x) = arctanh x = tanh−1 x |
∫ f (x) dx = x tanh −1 x + (1/2) ln (1 − x2) + c |
f (x) = arccsch x = csch−1 x |
∫ f (x) dx = x csch−1 x − sinh−1 x + c |
|
when x < 0 |
|
∫ f (x) dx = x csch−1 x + sinh−1 x + c |
|
when x > 0 |
f (x) = arcsech x = sech−1 x |
∫ f (x) dx = x sech−1 x − sin−1 x + c |
|
when sech−1 x < 0 |
|
∫ f (x) dx = x sech−1 x + sin−1 x + c |
|
when sech−1 x > 0 |
f (x) = arccoth x = coth−1 x |
∫ f (x) dx = x coth−1 x + (1/2) ln (x2 − 1) + c |
f (x) = arcsinh ax = sinh−1 ax |
∫ f (x) dx = x sinh−1 ax − (x2 + a−2)1/2 + c |
f (x) = arccosh ax = cosh−1 ax |
∫ f (x) dx = x cosh−1 ax + (x2 − a−2)1/2 + c |
|
when cosh−1 ax < 0 |
|
∫ f (x) dx = x cosh−1 ax − (x2 − a−2)1/2 + c |
|
when cosh−1 ax > 0 |
f (x) = arctanh ax = tanh−1 ax |
∫ f (x) dx = x tanh−1 ax + (1/2)a−1 ln (a−2 − x2) + c |
f (x) = arccsch ax = csch−1 ax |
∫ f (x) dx = x csch−1 ax − a−1 sinh−1 ax + c |
|
when x < 0 |
|
∫ f (x) dx = x csch−1 ax + a−1 sinh−1 ax + c |
|
when x > 0 |
f (x) = arcsech ax = sech−1 ax |
∫ f (x) dx = x sech−1 ax − a−1 sin−1 ax + c |
|
when sech−1 ax < 0 |
|
∫ f (x) dx = x sech−1 ax + a−1 sin−1 ax + c |
|
when sech−1 ax > 0 |
f (x) = arccoth ax = coth−1 ax |
∫ f (x) dx = x coth−1 ax + (1/2)a−1 ln (x2 − a−2) + c |