całeczki

[Novy]

\(\displaystyle{ \int\frac{(x^{2}-1)^{3}}{x}dx}\)


\(\displaystyle{ \int{xsin(2x^{2}+1)} dx}\)

\(\displaystyle{ \int\frac{x}{\sqrt{3-5x^{2}}}dx}\)


\(\displaystyle{ \int\frac{dx}{x^{4}+64}}\)


\(\displaystyle{ \int\frac{dx}{x\sqrt{x+9}}}\)

[mostostalek]

1. podnieś do potęgi i zapisz jako suma całek

2.\(\displaystyle{ t=2x^2+1}\)

3. \(\displaystyle{ x=\frac{\sqrt{3}}{\sqrt{5}}\sin{t}}\)

4.\(\displaystyle{ x=2\sqrt{2}\tan{t}}\)

[soku11]

5.
\(\displaystyle{ \int\frac{dx}{x\sqrt{x+9}}\\
x+9=t^2\\
x=t^2-9\\
dx=2tdt\\
2\int \frac{tdt}{(t^2-9)t}=
2\int \frac{dt}{(t-3)(t+3)}=
2(\int \frac{dt}{6(t-3)}-\int\frac{dt}{6(t+3)})=
\frac{1}{3}(\int \frac{dt}{t-3}-\int\frac{dt}{t+3})=\frac{1}{3}(ln|t-3|-ln|t+3|)+C=
\frac{1}{3}ln\left| \frac{t-3}{t+3}\right|+C=...}\)

POZDRO

[przemk20]

4 rozlóz na ulamki proste
\(\displaystyle{ x^4 + 64 = (x^4 + 16x^2 + 64) - 16 x^2 = (x^2 + 8)^2 - (4x)^2 = (x^2 - 4x +8)(x^2 +4x+8)}\)
3
\(\displaystyle{ 3-5x^2 = t, \ \ -10x dx = dt \iff xdx = -\frac{1}{10} dt \\
-\frac{1}{10} t \frac{dt}{\sqrt t}}\)