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\head {\bf Wyk�ad z analizy}\\
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Tydzie� 20.\\
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Metody ca�kowania
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\head �wiczenia
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\item{c20.1} Obliczy� ca�ki (a) $\int_0^1\frac{dx}{x^2+x+1},$
(b) $\int\frac{dx}{x^2-x+2},$
\newline (c) $\int_0^{\sqrt3}\frac{dx}{x^2+\sqrt3x+1},$
(d) $\int\frac{dx}{5x^2-2x+1}.$

\item{c20.2} Obliczy� ca�ki (a) $\int\frac{dx}{(2x^2+2x+1)^2},$
(b) $\int_0^{1/3}\frac{dx}{(9x^2+1)^4},$
\newline (c) $\int\frac{dx}{(x^2+\sqrt3x+1)^3}.$

\item{c20.3} Obliczy� ca�ki (a) $\int_0^{1/2}\frac{3x+2}{4x^2+1}\,dx,$
(b) $\int\frac{1-3x}{4x^2+3x+1}\, dx,$
\newline (c) $\int\frac{4x-3}{3x^2+3x+1}\,dx.$

\item{c20.4} Obliczy� ca�ki (a) $\int\frac{x^3+4x^2+5x-2}{x^4-1}\,dx,$
(b) $\int_0^{\sqrt3}\frac{x^3+x^2+x+2}{(x^2+1)(x^2+2)}\,dx,$
\newline (c) $\int_3^4\frac{5x^2-4x+4}{(x^2-4)^2}\,dx,$
(d) $\int_{\sqrt3}^3\frac{x^3+4x^2+7x+12}{(x^2+x)(x^2+3)}\,dx.$

\item{c20.5} Obliczy� ca�ki (a) $\int\frac{2e^{2x}+e^x+1}{e^{2x}+1}\,dx,$
(b) $\int\frac{1+\sinh x}{1+\cosh x}\,dx,$
\newline (c) $\int\frac{2x+3\sqrt{x+1}}{2x-3\sqrt{x+1}}\,dx,$
(d) $\int\frac{dx}{\sin x+\sqrt3\cos x},$
\newline (e) $\int\frac{dx}{9\cos^2x-16\sin^2x}.$

\head Zadania\endhead

\item{z20.1} Obliczy� $\int\cos^5 x\,dx$. (Wsk. Wyprowadzi� odpowiedni wz�r redukcyjny.)

\item{z20.2} Obliczy� $\int x^4\sin x\,dx$. (Wsk. Jak w z20.1.)

\item{z20.3} Obliczy� $\int\frac{\log^3x}{\sqrt x}\,dx$. (Wsk. Jak w z20.1.)

\item{z20.4} Obliczy� $\int_0^{1/\sqrt2}x^3\sqrt{1-x^2}\,dx.$

\item{z20.5} Obliczy� $\int\frac{dx}{(x^2+1)^{3/2}}.$





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