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Całka nieoznaczona. Całka oznaczona i jej zastosowania

\(\displaystyle{ \int_{}^{} (x ^{2} + 2sinx - \frac{1}{x})dx}\)

\(\displaystyle{ \int_{}^{} \frac{2 ^{x}-5 ^{x} }{10 ^{x} }dx}\)

a)
2x+2cosx-lnx

w czym wlasciwie problem ?
pierwsza rozbijasz na sume calek i liczysz z podstawowych wzorow

\(\displaystyle{ \int_{}^{} x ^{2}dx + \int_{}^{}2sinxdx - \int_{}^{}\frac{1}{x}dx}\)

druga zapisz w postaci

\(\displaystyle{ \int_{}^{} (\frac{2 ^{x}}{10 ^{x}} - \frac{5 ^{x} }{10 ^{x}})dx \Leftrightarrow\int_{}^{} (\frac{2 }{10})^{x}dx- \int_{}^{} (\frac{5 }{10})^{x}dx \Leftrightarrow \int_{}^{}(\frac{1 }{5})^{x}dx- \int_{}^{} (\frac{1}{2})^{x}dx}\)

Jefferson Darcy pisze:\(\displaystyle{ \int_{}^{} (x ^{2} + 2sinx - \frac{1}{x})dx}\)

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

Jefferson Darcy pisze:\(\displaystyle{ \int_{}^{} \frac{2 ^{x}-5 ^{x} }{10 ^{x} }dx}\)

\(\displaystyle{ \int \frac{2^x - 5^x}{10^x} dx = \int \frac{2^x}{10^x}dx - \int \frac{5^x}{10^x} dx = \int \left(\frac{1}{5}\right)^x dx - \int \left( \frac{1}{2} \right)^x dx = \frac{ \left( \frac{1}{5} \right)^x}{ln \frac{1}{5}} - \frac{ \left( \frac{1}{2} \right)^x}{ln \frac{1}{2}} + C}\)