randRange(2,9)
[
["\\sin(x)", "\\cos(x)", "\\sin(x)", "\\cos(x)", "\\sin(\\sin(x))", "\\cos(x)\\cos(\\sin(x))", ],
["\\sin(x)", "\\cos(x)", "\\cos(x)", "-\\sin(x)", "\\sin(\\cos(x))", "-\\cos(\\cos(x))\\sin(x)", ],
["\\sin(x)", "\\cos(x)", "e^x", "e^x", "\\sin(e^x)", "\\cos(e^x)e^x", ],
["\\sin(x)", "\\cos(x)", "x^"+n, n+"x^{"+(n-1)+"}", "\\sin(x^"+n+")", n+"x^{"+(n-1)+"}\\cos(x^"+n+")", ],
["\\sin(x)", "\\cos(x)", "1/x", "-1/x^2", "\\sin(1/x)", "-\\cos(1/x)/x^2", ],
["\\cos(x)", "-\\sin(x)", "\\sin(x)", "\\cos(x)", "\\cos(\\sin(x))", "-\\cos(x)\\sin(\\sin(x))", ],
["\\cos(x)", "-\\sin(x)", "\\cos(x)", "-\\sin(x)", "\\cos(\\cos(x))", "\\sin(x)\\sin(\\cos(x))", ],
["\\cos(x)", "-\\sin(x)", "e^x", "e^x", "\\cos(e^x)", "-e^x\\sin(e^x)", ],
["\\cos(x)", "-\\sin(x)", "x^"+n, n+"x^{"+(n-1)+"}", "\\cos(x^"+n+")", "-"+n+"x^{"+(n-1)+"}\\sin(x^"+n+")", ],
["\\cos(x)", "-\\sin(x)", "1/x", "-1/x^2", "\\cos(1/x)", "\\sin(1/x)/x^2", ],
["e^x", "e^x", "\\sin(x)", "\\cos(x)", "e^{\\sin(x)}", "\\cos(x)e^{\\sin(x)}", ],
["e^x", "e^x", "\\cos(x)", "-\\sin(x)", "e^{\\cos(x)}", "-e^{\\cos(x)}\\sin(x)", ],
["e^x", "e^x", "e^x", "e^x", "e^{e^x}", "e^x e^{e^x}", ],
["e^x", "e^x", "x^"+n, n+"x^{"+(n-1)+"}", "e^{x^"+n+"}", n+"x^{"+(n-1)+"}e^{x^"+n+"}", ],
["e^x", "e^x", "1/x", "-1/x^2", "e^{1/x}", "-e^{1/x}/x^2", ],
["x^"+n, n+"x^{"+(n-1)+"}", "\\sin(x)", "\\cos(x)", "\\sin^"+n+"(x)", n+"\\sin^{"+(n-1)+"}(x)\\cos(x)", ],
["x^"+n, n+"x^{"+(n-1)+"}", "\\cos(x)", "-\\sin(x)", "\\cos^"+n+"(x)", "-"+n+"\\cos^{"+(n-1)+"}(x)\\sin(x)", ],
["x^"+n, n+"x^{"+(n-1)+"}", "e^x", "e^x", "(e^x)^"+n, n+"(e^x)^{"+(n-1)+"}e^x", ],
]
randRange(0,functionBank.length-1)
functionBank[fNum]
Bestimmen Sie
\displaystyle \int f[5] dx.
Verwenden Sie C als Integrationskonstante.
f[4] + C
Bei der Integration durch Substitution verwenden wir die
Gleichung
\displaystyle \int f\left(g(x)\right)g'(x) dx =
F(g(x)) + C,
wobei F eine Stammfunktion von f ist.
Es geht also darum, Funktionen f und g
so zu identifizieren, dass
f\left(g(x)\right)g'(x) gleich dem Integranden
f[5] ist und wir eine
Stammfunktion F von f bestimmen können.
Hier eignen sich f(x) = f[1] und g(x)
= f[2] mit
g'(x)
= f[3].
Es sind dann F(x) = f[0] eine Stammfunktion
von f und
F(g(x)) = f[4].
Damit ergibt sich
\displaystyle \int f[5] dx =
f[4] + C.