Oblicz przez podstawienie calka nieoznaczona

[simpsone]

\(\displaystyle{ ∫ (\cos ^4(x)+ \sin ^8(x))dx}\)

[a4karo]

Ta całka jest równa `{x^2}/2(\cos 4+\sin 8)`.
Chyba, że chodziło Ci o `\int\cos(4x)+\sin(8x) dx` Wtedy możesz podstawić `t=4x` w pierwszej i `t=8x` w drugiej

[simpsone]

źle mi latex to przetlumaczyl poprawie to i spojrz teraz

Dodano po 30 sekundach:
cos i sinusy sa do kwadratu

Dodano po 1 minucie 39 sekundach:
cos i sinusy sa do kwadratu

[a4karo]

??? do kwadratu, czy do czwartej i ósmej potęgi?

I nie zwalaj na Latexa. To Ty źle wpisałeś

[simpsone]

\(\displaystyle{ ∫ \cos ^4(x)+ \sin ^8(x)dx}\) tak wyglada zadanie..
po skopiowaniu usunely mi sie ^^^

[a4karo]

Wyraź kwadrat kosinus i kwadrat sinusa przez np. Sinus podwojonego kąta. Potem podnieś to do odpowiednich potęg i zastosuj ten sam trick jeszcze kilka razy.

[Mariusz M]

\(\displaystyle{ \int_{}^{} \cos ^4(x)+ \sin ^8(x)dx}\)

Jedynka trygonometryczna i całkowanie przez części

\(\displaystyle{ \int_{}^{} \left( 1-\sin^{2}\left( x\right) \right)^2 + \sin ^8(x)dx\\
\int{1-2\sin^{2}{\left( x\right) }+\sin^{4}{\left( x\right) }+\sin^{8}{\left( x\right) } \dd x }\\
}\)

Wyprowadźmy wzór redukcyjny dla
\(\displaystyle{ \int{\sin^{n}\left( x\right) \dd x }}\)

\(\displaystyle{ \int{\sin^{n}{\left( x\right) } \dd x }=\int{\sin{\left( x\right) \cdot \sin^{n-1}{\left( x\right) } } \dd x }\\
\int{\sin^{n}{\left( x\right) } \dd x }=-\cos{\left( x\right) }\sin^{n-1}{\left( x\right) }-\int{\left( -\cos{\left( x\right) }\right)\left( \left( n-1\right)\sin^{n-2}\left( x\right)\cos{\left( x\right) } \right) \dd x }\\
\int{\sin^{n}{\left( x\right) } \dd x }=-\cos{\left( x\right) }\sin^{n-1}{\left( x\right) }+\left( n-1\right) \int{\sin^{n-2}{\left( x\right) }\cos^{2}\left( x\right) \dd x }\\
\int{\sin^{n}{\left( x\right) } \dd x }=-\cos{\left( x\right) }\sin^{n-1}{\left( x\right) }+\left( n-1\right)\int{\sin^{n-2}{\left( x\right) }\left( 1-\sin^{2}{\left( x\right) }\right) \dd x }\\
\int{\sin^{n}{\left( x\right) } \dd x }=-\cos{\left( x\right) }\sin^{n-1}{\left( x\right) }+\left( n-1\right)\int{\sin^{n-2}{\left( x\right) } \dd x }-\left( n-1\right)\int{\sin^{n}{\left( x\right) } \dd x }\\
n\int{\sin^{n}{\left( x\right) } \dd x }=-\cos{\left( x\right) }\sin^{n-1}{\left( x\right) }+\left( n-1\right)\int{\sin^{n-2}{\left( x\right) } \dd x }\\
\int{\sin^{n}{\left( x\right) } \dd x }=- \frac{1}{n} \cos{\left( x\right) }\sin^{n-1}{\left( x\right) }+\frac{n-1}{n}\int{\sin^{n-2}{\left( x\right) } \dd x }\\
}\)

\(\displaystyle{
\int{\sin^{8}{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }+\frac{7}{8}\left( -\frac{1}{6}\cos{\left( x\right) }\sin^{5}{\left( x\right) }+\frac{5}{6}\int{\sin^{4}{\left( x\right) } \dd x }\right)\\
\int{\sin^{8}{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}+\frac{35}{48}\int{\sin^{4}{\left( x\right) } \dd x }\\
\int{\sin^{8}{\left( x\right) } \dd x }+ \int{\sin^{4}{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}{\left( x\right) }+\frac{83}{48}\int{\sin^{4}{\left( x\right) } \dd x }\\
\int{\sin^{8}{\left( x\right) } \dd x }+ \int{\sin^{4}{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}{\left( x\right) }+\frac{83}{48}\left( -\frac{1}{4}\cos{\left( x\right) }\sin^{3}{\left( x\right) }+\frac{3}{4} \int{\sin^{2}{\left( x\right) } \dd x }\right)\\
\int{\sin^{8}{\left( x\right) } \dd x }+ \int{\sin^{4}{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}{\left( x\right) }-\frac{83}{192}\cos{\left( x\right) }\sin^{3}{\left( x\right) }+\frac{249}{192}\int{\sin^{2}{\left( x\right) } \dd x }\\
\int{\sin^{8}{\left( x\right) } \dd x }+ \int{\sin^{4}{\left( x\right) } \dd x }-2\int{\sin^2{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}{\left( x\right) }-\frac{83}{192}\cos{\left( x\right) }\sin^{3}{\left( x\right) }-\frac{45}{64}\int{\sin^{2}{\left( x\right) } \dd x }\\
\int{\sin^{8}{\left( x\right) } \dd x }+ \int{\sin^{4}{\left( x\right) } \dd x }-2\int{\sin^2{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}{\left( x\right) }-\frac{83}{192}\cos{\left( x\right) }\sin^{3}{\left( x\right) }-\frac{45}{64}\left( -\frac{1}{2}\cos{\left( x\right) }\sin{\left( x\right) }+\frac{1}{2}\int{ \dd x }\right) \\
\int{\sin^{8}{\left( x\right) } \dd x }+ \int{\sin^{4}{\left( x\right) } \dd x }-2\int{\sin^2{\left( x\right) } \dd x }=-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}{\left( x\right) }-\frac{83}{192}\cos{\left( x\right) }\sin^{3}{\left( x\right) }+\frac{45}{128}\cos{\left( x\right) }\sin{\left( x\right) }-\frac{45}{128}\int{ \dd x }\\
\int{\sin^{8}{\left( x\right) } \dd x }+ \int{\sin^{4}{\left( x\right) } \dd x }-2\int{\sin^2{\left( x\right) } \dd x }+\int{ \dd x }=\\
-\frac{1}{8}\cos{\left( x\right) }\sin^{7}{\left( x\right) }-\frac{7}{48}\cos{\left( x\right) }\sin^{5}{\left( x\right) }-\frac{83}{192}\cos{\left( x\right) }\sin^{3}{\left( x\right) }+\frac{45}{128}\cos{\left( x\right) }\sin{\left( x\right) }+\frac{83}{128}x+C
}\)