pare calek nieoznaczonych

[wegian]

Prosze o pomoc
1.\(\displaystyle{ \int_{}^{} \frac{2x ^{2} }{x ^{2}+9 } dx}\)
2. \(\displaystyle{ \int_{}^{} \frac{x ^{2}+1 }{x-1}dx}\)
3.\(\displaystyle{ \int_{}^{} \frac{x ^{2} }{6-x ^{6} }dx}\)
4.\(\displaystyle{ \int_{}^{} x ^{3} \sqrt{4-x ^{2} } dx}}\)

[tometomek91]

1.
\(\displaystyle{ \int \frac{2x ^{2}}{x ^{2}+9 } dx=\int \frac{2x ^{2}+18}{x ^{2}+9 } dx-18\int \frac{dx}{x ^{2}+9 }=2\int dx-18\int \frac{dx}{x ^{2}+9 }\\
\\
\int \frac{dx}{x ^{2}+9 }=\\
3u=x\\
3du=dx\\
=\int \frac{3du}{9u^2+9 }=\frac{1}{3}\int \frac{du}{u^2+1 }=\frac{1}{3}arctgu+C=\frac{1}{3}arctg\left(\frac{1}{3}x \right)+C\\
\\
2\int dx-18\int \frac{dx}{x ^{2}+9 }=2x-6arctg\left(\frac{1}{3}x \right)+C}\)

-- 5 wrz 2010, o 12:50 --

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

[jarek4700]

Trzecią całkę przedstaw jako:

\(\displaystyle{ \frac{1}{6}\int{}{}\frac{x^{2}}{1-(\frac{1}{\sqrt{6}}x^{3})^{2}}dx}\)

Następnie podstaw \(\displaystyle{ \frac{1}{\sqrt{6}}x^{3} = t}\) i rozbij to na ułamki proste.

[gott314]

4. \(\displaystyle{ \int_{}^{} x ^{3} \sqrt{4-x ^{2} } dx}=\left|\begin{array}{ccc} t=\sqrt{4-x^2}\\ x^2=4-t^2\\ -t\cdot \mbox{d}t=x\cdot \mbox{d}x \end{array}\right|=-\int (4-t^2)\cdot t \cdot t \ \mbox{d}t=}\)
\(\displaystyle{ =-4\int t^2 \ \mbox{d}t+\int t^4 \ \mbox{d}t=-\frac{4}{3}t^3+\frac{1}{5}t^5 +C=-\frac{4}{3}(4-x^2)^{\frac{3}{2}}+\frac{1}{5}(4-x^2)^{\frac{5}{2}} +C}\)

[wegian]

Wielkie dzieki !