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inactive; Cclosed; preserveAspect; startGroup; ] Supersin :[font = input; preserveAspect; ] SuperSin[x_] := Integrate[Sin[Sin[t]], {t, 0, x}] :[font = input; preserveAspect; startGroup; ] Plot[Evaluate@SuperSin[x], {x, -3Pi, 3Pi}] //Timing :[font = message; inactive; preserveAspect; ] NIntegrate::ploss: Numerical integration stopping due to loss of precision. Achieved neither the requested PrecisionGoal nor AccuracyGoal; suspect one of the following: highly oscillatory integrand or the true value of the integral is 0. :[font = message; inactive; preserveAspect; ] 1 Power::infy: Infinite expression -- encountered. 0. :[font = message; inactive; preserveAspect; ] NIntegrate::ploss: Numerical integration stopping due to loss of precision. 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = input; preserveAspect; startGroup; ] f[x_] := x^5 Exp[a x] Integrate[f[x], x] :[font = output; output; inactive; preserveAspect; endGroup; ] E^(a*x)*(-120/a^6 + (120*x)/a^5 - (60*x^2)/a^4 + (20*x^3)/a^3 - (5*x^4)/a^2 + x^5/a) ;[o] 2 3 4 5 a x -120 120 x 60 x 20 x 5 x x E (---- + ----- - ----- + ----- - ---- + --) 6 5 4 3 2 a a a a a a :[font = input; preserveAspect; startGroup; ] f[x_] := x^3 Cos[a x] Integrate[f[x], x] :[font = output; output; inactive; preserveAspect; endGroup; ] (3*(-2 + a^2*x^2)*Cos[a*x])/a^4 + (x*(-6 + a^2*x^2)*Sin[a*x])/a^3 ;[o] 2 2 2 2 3 (-2 + a x ) Cos[a x] x (-6 + a x ) Sin[a x] ----------------------- + ----------------------- 4 3 a a :[font = input; preserveAspect; startGroup; ] f[x_] := x^3 Sin[a x] Integrate[f[x], x] :[font = output; output; inactive; preserveAspect; endGroup; ] (x*(6 - a^2*x^2)*Cos[a*x])/a^3 + (3*(-2 + a^2*x^2)*Sin[a*x])/a^4 ;[o] 2 2 2 2 x (6 - a x ) Cos[a x] 3 (-2 + a x ) Sin[a x] ---------------------- + ----------------------- 3 4 a a :[font = input; preserveAspect; startGroup; ] f[x_] := x^3 Exp[I a x] ans = Integrate[f[x], x] :[font = output; output; inactive; preserveAspect; endGroup; ] E^(I*a*x)*(-6/a^4 + (6*I*x)/a^3 + (3*x^2)/a^2 - (I*x^3)/a) ;[o] 2 3 I a x -6 6 I x 3 x I x E (-- + ----- + ---- - ----) 4 3 2 a a a a :[font = input; preserveAspect; startGroup; ] cosint = Re[ans] //ComplexExpand sinint = Im[ans] //ComplexExpand :[font = output; output; inactive; preserveAspect; ] (-6/a^4 + (3*x^2)/a^2)*Cos[a*x] + ((-6*x)/a^3 + x^3/a)*Sin[a*x] ;[o] 2 3 -6 3 x -6 x x (-- + ----) Cos[a x] + (---- + --) Sin[a x] 4 2 3 a a a a :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup; ] ((6*x)/a^3 - x^3/a)*Cos[a*x] + (-6/a^4 + (3*x^2)/a^2)*Sin[a*x] ;[o] 3 2 6 x x -6 3 x (--- - --) Cos[a x] + (-- + ----) Sin[a x] 3 a 4 2 a a a ^*)