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\begin{document}
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\center{\Large 11. MATEMATIKA A1 FELADATSOR}

\begin{enumerate}
\item Határozza meg az alábbi integrálokat parciális integrálással:
\begin{enumerate}
\item $\int x\cos x dx$
\megold{$x\sin x+\cos x+c$}
\item $\int (3x+2)\sin x dx$
\megold{$-(2 + 3 x) \cos(x) + 3 \sin(x)+c$}
\item $\int x^2\cos 2x dx$
\megold{$\frac{1}{2}x^2 \sin(2 x) - \frac{1}{4} \sin(2 x) + \frac{1}{2} x \cos(2 x) + c$}
\item $\int (5x+3)e^x dx$
\megold{$5 e^x x - 2 e^x + c$}
\item $\int \frac{6x}{e^x} dx$
\megold{$-6 e^{-x} x - 6 e^{-x}+c$}
\item $\int (x^2+2x+3)e^{4x} dx$
\megold{$\frac{1}{4} e^{4 x} x^2+ \frac{3}{8} e^{4 x} x + \frac{21}{32} e^{4 x}+c $}
\item $\int x\ln x dx$
\megold{$-\frac{1}{4} x^2 + \frac{1}{2} x^2\ln x + c$}
\item $\int \frac{\ln x}{x^2} dx$
\megold{$-\frac{1}{x} - \frac{\ln x}{x}+c$}
\item $\int \ln x dx$
\megold{$x\ln x - x+c$}
\item $\int arctg x dx$
\megold{$x arctg x - \frac{1}{2}\ln (1 + x^2) + c$}
\item $\int arcsin x dx$
\megold{$\sqrt{1-x^2} + x arcsin x + c$}
\end{enumerate}
\item Határozza meg az alábbi integrálokat:
\begin{enumerate}
\item $\int \frac{6}{x+3} dx$
\megold{$6\ln |x+3|+c$}
\item $\int \frac{8}{3x-2} dx$
\megold{$\frac{8}{3}\ln |3x-2|+c$}
\item $\int \frac{10}{4-5x} dx$
\megold{$-2\ln |4-5x|+c$}
\item $\int \frac{5x+2}{x-2} dx$
\megold{$5 x + 12 \ln(x - 2) +c$}
\item $\int \frac{x^2+3x-1}{x-1} dx$
\megold{$\frac{1}{2} x^2+4x + 3 \ln(x-1) + c$}
\item $\int \frac{2x^3+4x^2+3x-7}{x-1} dx$
\megold{$\frac{2}{3} x^3 + 3 x^2 + 9 x + 2 \ln |x - 1| + c$}
\item $\int \frac{3}{1+16x^2} dx$
\megold{$\frac{3}{4}arctg (4x)+c$}
\item $\int \frac{1}{x^2+2x+2} dx$
\megold{$arctg(x+1)+c$}
\item $\int \frac{6}{4x^2+4x+2} dx$
\megold{$3arctg(2x+1)+c$}
\item $\int \frac{16x+5}{1+4x^2} dx$
\megold{$2\ln (1+4x^2)+\frac{5}{2}arctg (2x)+c$}
\item $\int \frac{6x-4}{x^2-2x+2} dx$
\megold{$3 \ln(x^2-2x+2) + 2 arctg (x-1) + c$}
\item $\int \frac{36x+28}{9x^2+6x+2} dx$
\megold{$2\ln (9 x^2 + 6 x + 2) + \frac{16}{3} arctg(3 x + 1)) + c$}
\item $\int \frac{28x^3+48x^2-12x+47}{1+4x^2} dx$
\megold{$\frac{7}{2}x^2 + 12 x + \frac{35}{2} arctg(2 x) - \frac{19}{8} \ln(4 x^2 + 1) + c$}
\item $\int \frac{8}{x^2-4} dx$
\megold{$2\ln |x-2| - 2\ln|x + 2| + c$}
\item $\int \frac{9}{x^2+x-6} dx$
\megold{$\frac{9}{5} \ln |x-2| -\frac{9}{5} \ln |x+3| + c$}
\item $\int \frac{7}{x^2+2x-3} dx$
\megold{$\frac{7}{4} \ln |x-1| -\frac{7}{4} \ln |x+3| + c$}
\item $\int \frac{14x-2}{x^2-x-12} dx$
\megold{$\frac{54}{7} \ln |x-4| + \frac{44}{7} \ln |x + 3| + c$}
\item $\int \frac{3x^2-5x+9}{x^2+3x-4} dx$
\megold{$3x + \frac{7}{5} \ln |x-1| - 77 \ln |x + 4| + c$}
\item $\int \frac{5x^3+6x^2-7x+9}{x^2+6x+8} dx$
\megold{$\frac{5}{2} x^2 -24x+ \frac{7}{2} \ln |x+2| + \frac{187}{2} \ln |x+4| + c$}
\end{enumerate}
\end{enumerate}


\end{document} 