Indefinite Integral

Author

Eric W. Weisstein
April 11, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/IndefiniteIntegral.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/IndefiniteIntegral.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Mathematica Can Do

∫((∂(x (Sqrt[sin(x)])))/(∂x))dx

v4.2

Integrate[D[x Sqrt[Sin[x]],x],x]

v5

Integrate[D[x Sqrt[Sin[x]],x],x]

v6

Integrate[D[x Sqrt[Sin[x]],x],x]

∫(x^x)(log(x) +1)dx

v5.2

Integrate[(x^x)(Log[x]+1),x]

v6

Integrate[(x^x)(Log[x]+1),x]

Mathematica Cannot Do

∫((∂(Subscript[Li, 2])(x log(x)))/(∂x))dx

v4.2

Integrate[D[PolyLog[2,x Log[x]],x],x]

v5.0

Integrate[D[PolyLog[2,x Log[x]],x],x]

v6.0.1

Integrate[D[PolyLog[2,x Log[x]],x],x]

∫(∂/(∂x))((e^(-(cos^(-1))(x)))/(Sqrt[1-(x^2)]))dx

Check

FullSimplify[((((E^(-ArcCos[x])) x)/(((1-(x^2)))^(3/2)))+((E^(-ArcCos[x]))/(1-(x^2))))]

Simplify[(E^(-ArcCos[x])) (((x+(Sqrt[1-(x^2)]))/(((1-(x^2)))^(3/2))))-(E^(-ArcCos[x])) ((x/(((1-(x^2)))^(3/2)))+(1/(1-(x^2))))]

v6

Integrate[D[((E^(-ArcCos[x]))/(Sqrt[1-(x^2)])),x],x]

∫(∂/(∂x))((1/2) erf(a x) erf(b x) (Sqrt[π]))dx

v6

Integrate[D[ Erf[a x] Erf[b x] Sqrt[Pi]/2, x], x]

∫((5 (x^9)-5 (x^6)+9 (x^5)-5 (x^4)-(x^2)+1)/(((x^5)-(x^2)+x-1) (Sqrt[(x^5)+x-1])))dx

expr=(5 x^9 - 5 x^6 + 9 x^5 - 5 x^4 - x^2 + 1)/ ((x^5 - x^2 + x - 1) Sqrt[x^5 + x - 1]);

Check

expr2=D[-Log[(-2x Sqrt[x^5+x-1]-x^5-x^2-x+1)/(x^5-x^2+x-1)]+2 Sqrt[x^5+x-1],x]//Simplify

expr-expr2//Simplify

v6

Integrate[expr, x]

∫(x^x) (Sqrt[x]) (e^(-x))dx

Cannot be done in closed form?

v6

Integrate[(x^x)(Sqrt[x])Exp[-x],x]

Irreducible

∫(e^(-(x^2)))dx

Integrate[Exp[-x^2],x]

∫((e^(-x))/x)dx

Integrate[((Exp[-x])/x),x]

∫((sin(x))/x)dx

Integrate[((Sin[x])/x),x]

∫((cos(x))/x)dx

Integrate[((Cos[x])/x),x]

∫(1/(log(x)))dx

Integrate[(1/(Log[x])),x]

∫(x^x)dx

Integrate[(x^x),x]

∫(x^(-x))dx

Integrate[(x^(-x)),x]

∫(Sqrt[sin(x)])dx

Integrate[(Sqrt[Sin[x]]),x]

D[%,x]//FullSimplify

∫x(Sqrt[sin(x)])dx

Integrate[x(Sqrt[Sin[x]]),x]

D[%,x]//FullSimplify

∫(x^2)(Sqrt[sin(x)])dx

Integrate[x^2(Sqrt[Sin[x]]),x]

∫(x^3)(Sqrt[sin(x)])dx

Integrate[x^3(Sqrt[Sin[x]]),x]

∫(Sqrt[cos(x)])dx==2 E((x/2)|2)

Integrate[Sqrt[Cos[x]],x]

D[%,x]//Simplify

∫(Sqrt[log(x)])dx

Integrate[Sqrt[Log[x]],x]//FullSimplify

D[%,x]//FullSimplify

∫(e^(e^x))dx==Ei((e^x))

Integrate[Exp[Exp[x]],x]//FullSimplify

D[%,x]//Simplify

∫log(log(x))dx==x log(log(x))-li(x)

Integrate[Log[Log[x]],x]//FullSimplify

D[%,x]//Simplify

Constant of Integration

∫(((x+1))^10)dx-∫Expand[(((x+1))^10)]dx

Expand[(((x+1))^10)]

Table[Integrate[(1+x)^n,x]-Integrate[Expand[(1+x)^n],x]//Simplify,
{n,10}]

Common Integrals

Powers

(x^r)

Integrate[(x^r),x]

(1/x)

Integrate[1/x,x]

Integrate[1/x,x,Assumptions->{x<0}]

(a^x)

Integrate[(a^x),x]

Trigonometric

sin x

Integrate[Sin[x],x]

cos x

Integrate[Cos[x],x]

tan x

Integrate[Tan[x],x]

csc x

Mathematica's solution

i=Integrate[Csc[x],x]

My standard solution

D[Csc[x]-Cot[x],x]

My solutions

FullSimplify[D[#,x]]&/@{
Log[Csc[x]-Cot[x]],
Log[Tan[x/2]],
Log[(1-Cos[x])/(1+Cos[x])]/2
}

Show[GraphicsArray[Partition[Plot[#,{x,-10,10},PlotPoints->100,DisplayFunction->Identity]&/@{Log[Csc[x]-Cot[x]],
Log[Tan[x/2]],
Log[(1-Cos[x])/(1+Cos[x])]/2,
-Log[Cos[(x/2)]]+Log[Sin[(x/2)]]},
{2}]]]

sec x

Integrate[Sec[z],z]

Simplify[D[%,z]]

Simplify[D[Log[Sec[z]+Tan[z]],z]]

cot x

Integrate[Cot[z],z]

Combinations of trigs

Integrate[Sin[z]^2,z]

Integrate[Cos[z]^2,z]

Integrate[Tan[z]^2,z]

Integrate[Sec[z]^2,z]

Integrate[Csc[z]^2,z]

Integrate[Cot[z]^2,z]

Integrate[Sec[z]Tan[z],z]

Inverse Trig

Integrate[ArcCos[z],z]

Integrate[ArcSin[z],z]

Integrate[ArcTan[z],z]

Rational and square roots

Integrate[1/Sqrt[a^2-z^2],z]

Simplify[%]

Simplify[D[%,z]]

D[ArcSin[z/a],z]

FullSimplify[D[ArcSin[z/a],z],a>0]

Jacobi

PowerExpand[Integrate[JacobiSN[u,k^2],u]]

PowerExpand[Integrate[JacobiCN[u,k^2],u]]

PowerExpand[Integrate[JacobiDN[u,k^2],u]]

Jacobi squares

FullSimplify[Integrate[JacobiSN[u,k^2]^2,u]]

(sin^n)x and (cos^n)x

SinReduce[l_?EvenQ]:=Module[{n=l/2},
  1/2^(2n) Binomial[2n,n]+(-1)^n/2^(2n-1)*
  Sum[(-1)^k Binomial[2n,k]Cos[2(n-k)x],{k,0,n-1}]//Together
]

SinReduce[l_?OddQ]:=Module[{n=(l-1)/2},
  (-1)^n/4^n Sum[(-1)^k Binomial[2n+1,k]Sin[(2n+1-2k)x],{k,0,n}]//Together
]

CosReduce[l_?EvenQ]:=Module[{n=l/2},
  (-1)^n/2^(2n) Binomial[2n,n]+1/2^(2n-1)*
  Sum[Binomial[2n,k]Cos[2(n-k)x],{k,0,n-1}]//Together
]

CosReduce[l_?OddQ]:=Module[{n=(l-1)/2},
  1/4^n Sum[Binomial[2n+1,k]Cos[(2n+1-2k)x],{k,0,n}]//Together
]

CosReduce[8]

SinReduce[8]

Simplify[%]

Unprotect[Power];
0^0=1;
Protect[Power];
SincIntegrate[a_,b_]:=Module[
  {c=Mod[a-b,2]},
  Pi^(1-c)(-1)^Floor[(a-b)/2]/2^(a-c)/(b-1)! Sum[(-1)^k Binomial[a,k]*
  (a-2k)^(b-1)Log[a-2k]^c,{k,0,Floor[a/2]-c}]
]

SincIntegrate[5,5]

Integrate[Sin[x]^3/x^3,{x,0,Infinity}]

EIntegrate[Cos[x_]^m_?EvenQ,x_]:=Module[{n=m/2},
  Expand[(2n-1)!!/(2n)!!(Sin[x]Sum[(2k)!!/(2k+1)!! Cos[x]^(2k+1),{k,0,n-1}]+x)]
]
EIntegrate[Cos[x_]^m_?OddQ,x_]:=Module[{n=(m-1)/2},
  Expand[(2n)!!/(2n+1)!! Sin[x]Sum[(2k-1)!!/(2k)!! Cos[x]^(2k),{k,0,n}]]
]
EIntegrate[Cos[x_]^m_,{x_,a_,b_}]:=Module[{c=CosInt[m]},
  (c/.x->b) - (c/.x->a)
]

Integrate[Cos[x]^5,x]

Simplify[D[%4,x]]

EIntegrate[Cos[x]^5,x]

Simplify[D[%8,x]]