Функции и интеграли

[Гост]

Задачи за функции и интеграли

[aifC]

Зад. 1
[tex]f(x) = \frac{arccos(x)}{sin(x)+cos(x)} \Rightarrow[/tex]
[tex]\frac{arccos(0)}{sin(0)+cos(0)} + \frac{\frac{d}{dx}\left(\frac{arccos(x)}{sin(x)+cos(x)}\right)}{1!} \cdot x + \frac{\frac{d^{2}}{dx^{2}}\left(\frac{arccos(x)}{sin(x)+cos(x)}\right)}{2!} \cdot x^{2} + \frac{\frac{d^{3}}{dx^{3}}\left(\frac{arccos(x)}{sin(x)+cos(x)}\right)}{3!} \cdot x^{3} + ......[/tex]
[tex]= \frac{\pi}{2} - \frac{(\pi + 2)x}{2} + \frac{(3\pi + 4)x^{2}}{4} - \frac{(11\pi + 20)x^{3}}{12} + .......[/tex]

[aifC]

Зад. 5
а) [tex]\int \frac{x^{2}-3}{(x^2+x+1)^{2}}dx = -\frac{4}{3}\int \frac{1}{x^2+x+1}dx - \frac{7x+2}{3(x^{2}+x+1)} = \int \frac{1}{\left(x+\frac{1}{2}\right)^{2}+\frac{3}{4}}dx;[/tex]
[tex]u = \frac{2x+1}{\sqrt{3}} \rightarrow dx = \frac{\sqrt{3}}{2}du; \Rightarrow \frac{2}{\sqrt{3}} \int \frac{1}{u^{2}+1}du = arctg(u) = -\frac{8arctg \left(\frac{2x+1}{\sqrt{3}}\right)}{3^{\frac{3}{2}}}+\frac{7x+2}{(x^{2}+x+1)}+C;[/tex]

б)[tex]\int \frac{1}{cos^{3}(x)}dx = \int sec^{3}dx = \frac{1}{2}\int sec(x)dx + \frac{sec(x) \cdot tg(x)}{2} = ln(tg(x)+sec(x)) = \frac{ln(|tg(x)+sec(x)|)+sec(x) \cdot tg(x)}{2} + C;[/tex]

в)[tex]\int \frac{1}{1-sin^{4}(x)}dx = -\int \frac{1}{sin^{4}(x)-1}dx = \int sec^{2}(x) \left(-\frac{tg^{2}(x)+1}{2tg^{2}(x)+1}\right)dx; u = tg(x) \rightarrow \frac{1}{sec^{2}(x)}du[/tex]
[tex]\frac{u^{2}+1}{2u^{2}+1}du = \frac{1}{2}\int \frac{1}{2u^{2}+1}du + \frac{1}{2}\int 1du = arctg(v)+u = \frac{arctg(\sqrt{2}tg(x))}{2^{\frac{3}{2}}} + \frac{tg(x)}{2} + C;[/tex]

[aifC]

Зад. 4
б) [tex]ch^{2}(x) - sh^{2}(x) = 1 \Rightarrow sh^{2}(x) + 1 = ch^{2}(x)[/tex]
[tex]\int \sqrt{x^{2}+1}dx \Rightarrow x = sh(u) \rightarrow u = arsh(x),dx = ch(u)du; \Rightarrow \int ch(u)\sqrt{sh^{2}(u)+1}du = \int ch^{2}(u)du = \frac{ch(u)sh(u)}{2} + \frac{1}{2} \int 1 du = \frac{arsh(x)+x\sqrt{x^{2}+1}}{2} + C;[/tex]

в) [tex]x = tg(u) \rightarrow u = arctg(x), dx = sec^{2}du;[/tex]
[tex]\int sec^{2}(u)\sqrt{tg^{2}(u)+1}du = \int sec^{3}du = \frac{1}{2}\int sec(u) du + \frac{sec(u)tg(u)}{2} = ln(sec(u)+tg(u)) = \frac{ln(|\sqrt{x^{2}+1}+x|)+x\sqrt{x^{2}+1}}{2} + C;[/tex]

[aifC]

Зад. 2
а) [tex]f(x) = x \cdot arcsin \left(\frac{2x}{1+x^2}\right) \Rightarrow f'(x) = arcsin \left(\frac{2x}{1+x^2}\right) + \frac{2x\sqrt{(x^{2}+1)^{2}}(1-x^{2})}{\sqrt{(x+1)^{2}}\sqrt{(x-1)^{2}}(1+x^{2})^{2}}[/tex]
ДМ:[tex]x \in (-\infty,\infty);[/tex]
Асимптоти няма,не е периодична.
Пресича [tex]Y - (0,0);[/tex]
[tex]f(-x) = f(x) \Rightarrow[/tex] Функцията е четна.
Графика:

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