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\begin{document}
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\center{\Large 12. MATEMATIKA A1 FELADATSOR}

\begin{enumerate}
\item Határozza meg az alábbi integrálokat:
\begin{enumerate}
\item $\int \cos^2x dx$
\megold{$\frac{x}{2}+\frac{\sin 2x}{4}+c$}
\item $\int \sin^23x dx$
\megold{$\frac{x}{2}-\frac{\sin 6x}{12}+c$}
\item $\int \sin x \cos 2x dx$
\megold{$frac{\cos x}{2}+\frac{\cos 3x}{6}+c$}
\item $\int \sin x \sin 2x dx$
\megold{$\frac{\sin x}{2}+\frac{\sin 3x}{6}+c$}
\item $\int \cos 2x \cos 3x dx$
\megold{$\frac{\sin x}{2}+\frac{\sin 5x}{10}+c$}
\item $\int \sin^3x dx$
\megold{$-\frac{3}{4}\cos x+\frac{\cos 3x}{12}+c$}
\item $\int \cos ^3x dx$
\megold{$\frac{3}{4}\sin x+\frac{\sin 3x}{12}+c$}
\item $\int \sin^4x dx$
\megold{$\frac{3}{8}x-\frac{\sin 2x}{4}+\frac{\sin 4x}{32}+c$}
\item $\int \sin^2x\cos 3x dx$
\megold{$-\frac{\sin x}{4}+\frac{\sin 3x}{6}-\frac{\sin 5x}{20}+c$}
\item $\int \cos^24x \sin x dx$
\megold{$-\frac{\cos x}{2}+\frac{\cos 7x}{28}-\frac{\cos 9x}{36}+c$}
\item $\int \cos^23x \sin ^22x  dx$
\megold{$\frac{x}{4}-\frac{\sin 2x}{16}-\frac{\sin 4x}{16}+\frac{\sin 6x}{24}-\frac{\sin 10x}{80}+c$}
\end{enumerate}
\item Határozza meg az alábbi integrálokat helyettesítéssel:
\begin{enumerate}
\item $\int e^x\sin(e^x) dx$, $t=e^x$
\megold{$-\cos(e^x)+c$}
\item $\int e^{2x}\cos(e^x) dx$, $t=e^x$
\megold{$e^x\sin (e^x)+\cos (e^x)+c$}
\item $\int \frac{e^{2x}}{1+e^x} dx$, $t=e^x$
\megold{$e^x-\ln (1+e^x)+c$}
\item $\int \frac{e^x}{1+e^{2x}} dx$, $t=e^x$
\megold{$\ln |\arctg x|+c$}
\item $\int \frac{1}{e^x-1} dx$, $t=e^x$
\megold{$\ln |e^x-1|-x+c$}
\item $\int \frac{e^x}{e^{2x}-1} dx$, $t=e^x$
\megold{$\frac{ln|e^x-1|}{2}+\frac{\ln |e^x+1|}{2}+c$}
\item $\int \frac{\sin \sqrt{x}}{\sqrt{x}} dx$, $t=\sqrt{x}$
\megold{$-2\cos (\sqrt{x})+c$}
\item $\int \cos \sqrt{2x} dx$, $t=\sqrt{x}$
\megold{$\sqrt{2x}\sin (\sqrt{2x})+\cos (\sqrt{2x})+c$}
\item $\int \frac{1}{2x-1-\sqrt{2x-1}} dx$, $t=\sqrt{2x-1}$
\megold{$\ln |1-\sqrt{2x-1}|+c$}
\item $\int \frac{6}{1+\sqrt[3]{x}} dx$, $t=\sqrt[3]{x}$
\megold{$9x^{2/3}-18x^{1/3}+18\ln |x^{1/3}+1|+c$}
\item $\int e^{\sqrt{x}} dx$, $t=\sqrt{x}$
\megold{$2\sqrt{x}e^{\sqrt{x}}-2e^{\sqrt{x}}+c$}
\item $\int x\sqrt{3x-2} dx$, $t=\sqrt{3x-2}$
\megold{$\frac{2}{15}(3x-2)^{5/2}+\frac{4}{27}(3x-2)^{3/2}+c$}
\item $\int x^2\sqrt{x-1}$, $t=\sqrt{x-1}$
\megold{$\frac{2}{7}(x-1)^{7/2}+\frac{18}{35}(x-1)^{5/2}+\frac{2}{3}(x-1)^{3/2}+c$}
\end{enumerate}
\end{enumerate}


\end{document} 