Obliczyć całki nieoznaczone.

[faraus]

1) \(\displaystyle{ \int_{}^{} (arccosx) ^{2}}\) \(\displaystyle{ dx}\)
2) \(\displaystyle{ \int_{}^{} xsin ^{2} x}\) \(\displaystyle{ dx}\)
3) \(\displaystyle{ \int_{}^{} \frac{dx}{2+ \sqrt{x} }}\)
4) \(\displaystyle{ \int_{}^{} \frac{x ^{3}dx}{x+1}}\)
5) \(\displaystyle{ \int_{}^{} sin ^{3} x}\) \(\displaystyle{ dx}\)
6) \(\displaystyle{ \int_{}^{} \frac{dx}{ \sqrt{4x - x ^{2} } }}\)
7) \(\displaystyle{ \int_{}^{} \frac{x ^{3} dx}{(x-1) ^{100} }}\)

[fermat]

4) \(\displaystyle{ \int_{}^{} \frac{x ^{3}dx}{x+1}}\)

\(\displaystyle{ \int_{}^{} \frac{x ^{3}dx}{x+1} = \int \frac {x^{3} + 1 - 1}{x + 1}dx = \int \frac {(x + 1)(x^{2} - x + 1)}{x + 1}dx - \int \frac {1}{x + 1}dx = \frac {1}{3}x^{3} - \frac{1}{2}x^{2} + x - ln|x + 1| + C}\)

[sea_of_tears]

5)
\(\displaystyle{ \int \sin^3xdx=\int \sin x\cdot \sin^2xdx=
\int \sin x\cdot (1-\cos^2x)dx=
\\
\begin{cases}
t=\cos x \\
dt=-\sin x dx
\end{cases}
\\
=\int -(1-t^2)dt=
\int (-1+t^2)dt=
-t+\frac{t^3}{3}+c=
-\cos x+\frac{\cos^3x}{3}+c}\)

-- 20 lutego 2009, 20:08 --

3)
\(\displaystyle{ \int \frac{dx}{2+\sqrt{x}}=\\
\begin{cases}
t=\sqrt{x} \\
t^2=x \\
2tdt=dx
\end{cases}
\\
=\int \frac{2tdt}{2+t}=
2\int\frac{t}{2+t}dt=
2\int \frac{2+t-2}{2+t}dt=
2\int \frac{2+t}{2+t}dt - 4\int \frac{dt}{2+t}=
2t -4 \ln(2+t)+c=
2\sqrt{x}-4\ln(2+\sqrt{x})+c}\)

[fermat]

6) \(\displaystyle{ \int_{}^{} \frac{dx}{ \sqrt{4x - x ^{2} } }}\)

6) \(\displaystyle{ \int_{}^{} \frac{dx}{ \sqrt{4x - x ^{2} } } = \int \frac {d(x - 2)}{\sqrt{2^{2} - (x - 2)^{2}}} = arcsin \frac {(x - 2)}{2} + C}\)

-- 20 lut 2009, o 20:19 --

7) \(\displaystyle{ \int_{}^{} \frac{x ^{3} dx}{(x-1) ^{100} }}\)

\(\displaystyle{ x - 1 = t}\)
\(\displaystyle{ dx = dt}\)
\(\displaystyle{ x^{3} = (t + 1)^{3}}\)

7) \(\displaystyle{ \int_{}^{} \frac{x ^{3} dx}{(x-1) ^{100} } = \int \frac{t^{3} + 3t^{2} + 3t + 1}{t^{100}}dt = \int t^{-97}dt + 3 \int t^{-98}dt + 3\int t^{-99}dt + \int \frac{1}{t}dt = ...}\)

-- 20 lut 2009, o 20:51 --

1) \(\displaystyle{ \int_{}^{} (arccosx) ^{2}}\) \(\displaystyle{ dx}\)
2) \(\displaystyle{ \int_{}^{} xsin ^{2} x}\) \(\displaystyle{ dx}\)

pierwsze dwie całki przez części,

w pierwszej dwa razy przez części, wszystko się powinno poskracać, a w drugiej skorzystaj ze wzoru:

za \(\displaystyle{ u}\) weź \(\displaystyle{ x}\) żeby się go pozbyć, a do obliczenia funkcji pierwotnej \(\displaystyle{ \ \ \ dv = sin^{2}x}\) zastosuj \(\displaystyle{ sin^{2}x = \frac {1 - cos2x}{2}}\)