1.\(\displaystyle{ \int\frac{x^{2}}{5 - x^{6}}dx}\)
2.\(\displaystyle{ \int\frac{3 + x}{3 - x}dx}\)
3.\(\displaystyle{ \int\frac{dx}{x^{2}+ 2x +3}}\)
4.\(\displaystyle{ \int\cos(\ln (x))dx}\)
5.\(\displaystyle{ \int\ln^{2} (x)dx}\)
6.\(\displaystyle{ \int\ x\cdot 2^{-x}dx}\)
proszę o pomoc przy tych całeczkach...
całki nieoznaczone
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- Użytkownik
- Posty: 6607
- Rejestracja: 16 sty 2007, o 19:42
- Płeć: Mężczyzna
- Podziękował: 119 razy
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całki nieoznaczone
1.
\(\displaystyle{ x^6=5t^2\\
x^3=\sqrt{5}t\\
x^2dx=\frac{\sqrt{5}}{3}dt\\
\frac{\sqrt{5}}{3} t \frac{dt}{5-5t^2}=
\frac{\sqrt{5}}{15} t \frac{dt}{1-t^2}=...}\)
2.
\(\displaystyle{ \int\frac{3 + x}{3 - x}dx=-\int \frac{x+3}{x-3}dx=
-\int \frac{x-3+6}{x-3}dx=-\int dx-6\int \frac{dx}{x-3}=...}\)
3.
\(\displaystyle{ \int\frac{dx}{x^{2}+ 2x +3} =
t\frac{dx}{(x+ 1)^2 +2} \\
(x+1)^2=2t^2\\
x+1=\sqrt{2}t\\
dx=\sqrt{2}dt\\
\sqrt{2} t \frac{dt}{2t^2+2}=\frac{\sqrt{2}}{2} t \frac{dt}{t^2+1}=...}\)
4.
\(\displaystyle{ \int\cos(\ln (x))dx = t \frac{x\cos(\ln (x))}{x}dx\\
ln(x)=t\\
x=e^t\\
\frac{dx}{x}=dt\\
t e^t cost dt\\
u=e^t\quad dv=cost dt \\
...}\)
5.
\(\displaystyle{ \int\ln^{2} (x)dx\\
u=lnx\quad dv=lnxdx\\
du=\frac{dx}{x}\quad v=x(lnx-1)\\
lnx\cdot x(lnx-1)-\int lnx dx-\int dx=...}\)
6 przez czesci.
POZDRO
\(\displaystyle{ x^6=5t^2\\
x^3=\sqrt{5}t\\
x^2dx=\frac{\sqrt{5}}{3}dt\\
\frac{\sqrt{5}}{3} t \frac{dt}{5-5t^2}=
\frac{\sqrt{5}}{15} t \frac{dt}{1-t^2}=...}\)
2.
\(\displaystyle{ \int\frac{3 + x}{3 - x}dx=-\int \frac{x+3}{x-3}dx=
-\int \frac{x-3+6}{x-3}dx=-\int dx-6\int \frac{dx}{x-3}=...}\)
3.
\(\displaystyle{ \int\frac{dx}{x^{2}+ 2x +3} =
t\frac{dx}{(x+ 1)^2 +2} \\
(x+1)^2=2t^2\\
x+1=\sqrt{2}t\\
dx=\sqrt{2}dt\\
\sqrt{2} t \frac{dt}{2t^2+2}=\frac{\sqrt{2}}{2} t \frac{dt}{t^2+1}=...}\)
4.
\(\displaystyle{ \int\cos(\ln (x))dx = t \frac{x\cos(\ln (x))}{x}dx\\
ln(x)=t\\
x=e^t\\
\frac{dx}{x}=dt\\
t e^t cost dt\\
u=e^t\quad dv=cost dt \\
...}\)
5.
\(\displaystyle{ \int\ln^{2} (x)dx\\
u=lnx\quad dv=lnxdx\\
du=\frac{dx}{x}\quad v=x(lnx-1)\\
lnx\cdot x(lnx-1)-\int lnx dx-\int dx=...}\)
6 przez czesci.
POZDRO