Integrals, Sums and Products

Mathematica has unprecedented calculus capabilities. Here are a few of the integrals, sums and products that it can do.

Indefinite Integrals

∫(Sqrt[1 + x])(Sqrt[2 + x])\[DifferentialD]x

∫(Sqrt[x + (Sqrt[1 + x])])\[DifferentialD]x

∫(Sqrt[x + (Sqrt[x])]) \[DifferentialD]x // FullSimplify

∫ArcTan[(x^2)] \[DifferentialD]x // FullSimplify

∫(Sqrt[Tan])[x] \[DifferentialD]x // FullSimplify

∫(Sqrt[Log])[x] \[DifferentialD]x

∫Log[1 + x + (x^12)] \[DifferentialD]x

∫(Sqrt[(a + b x) (c + d x)]) 
        \[DifferentialD]x

∫(1/(1 + (x^5))) \[DifferentialD]x

∫(1/(Sqrt[1 + (x^5)])) \[DifferentialD]x

∫(1/(Sqrt[1 + (x^6)])) \[DifferentialD]x

∫(x^m) (((a + b x))^n) \[DifferentialD]x

∫Exp[(x^n)] \[DifferentialD]x

∫Tan[x]^n \[DifferentialD]x

∫Sin[Tan[x]] \[DifferentialD]x // FullSimplify

∫Tan[Log[x]] \[DifferentialD]x

FullSimplify[%]

∫ArcCsc[Sec[Sec[x]]] \[DifferentialD]x // FullSimplify

\(∫\(ArcSin[x]\^2\/x\) \[DifferentialD]x // FullSimplify\)

∫(a^((x^r))) (x^b) Log[c (x^s)] \[DifferentialD]x

∫Erf[(x^a)] \[DifferentialD]x

∫AiryAi[x] \[DifferentialD]x

∫(x^n) EllipticK[x] \[DifferentialD]x

∫ProductLog[x]^n \[DifferentialD]x

∫∫∫∫∫∫Exp[-(x^2)] \[DifferentialD]x \[DifferentialD]x 
                  \[DifferentialD]x \[DifferentialD]x \[DifferentialD]x 
                  \[DifferentialD]x

Definite Integrals

(∫_0^∞) Log[
            x] Exp[-(x^2)] \[DifferentialD]x

(∫_0^π) Log[1 + Sin[θ]] \[DifferentialD]θ // 
      FullSimplify

(∫_0^π) Sin[Sin[x]] \[DifferentialD]x

(∫_0^∞) Exp[-x] (Sqrt[1 + x]) 
        \[DifferentialD]x

(∫_0^∞) ((x^n)/(1 + Exp[x])) 
        \[DifferentialD]x

(∫_0^1)(((1 - x))^m) (x^n) Log[x] 
        \[DifferentialD]x

(∫_0^∞) BesselJ[0, x] BesselJ[1, x] BesselK[0, x] 
        \[DifferentialD]x

(∫_0^∞) BesselJ[0, x] BesselJ[1, x] BesselJ[2, x] 
        \[DifferentialD]x

Checking Integrals

To check an indefinite integral, you can differentiate the result you get.

∫(Sqrt[x]) Sin[x] \[DifferentialD]x

D[%, x]

∫((x^5)/(1 + Exp[x])) \[DifferentialD]x

D[%, x]

Simplify[%]

(∫_0^∞) Log[
            x] Exp[-(x^2)] \[DifferentialD]x

N[%]

NIntegrate[Log[x] Exp[-(x^2)], {x, 0, ∞}]

Sums and Products

(∑_(k = 1)^n) (1/(k^6))

(∑_(k = 1)^n) (1/(1 + (k^4)))

(∑_(k = 1)^n)(((Binomial[n, k]/k))^3) (x^k)

(∑_(k = 1)^∞) ((x^k)/(k! (k + α)!))

(∑_(k = 1)^∞) PolyGamma[k]/(k^4)

(∏_(k = 2)^n)(( (k^3) + 1)/((k^3) - 1))

(∏_(k = 1)^n) (1/((k^3) + 1))

(∏_(k = 2)^∞)(1 - (1/(k^6)))