☰ matematika2

Zadaci za vježbu NEODREĐENI INTEGRAL ODREĐENI INTEGRAL

Rješenja zadataka za vježbu

1.: $\displaystyle\frac{2}{15}\sqrt{x}\left( -15+25x+3x^{2}\right) +C$ .
2.: $\displaystyle\frac{2^{-x}}{5\ln 2}-2\frac{5^{-x}}{\ln 5}+C$ .
3.: $\displaystyle\frac{\mathop{\mathrm{arctg}}\nolimits \left( \frac{x}{a}\right) }{% a}+C$ .
4.: $\displaystyle\mathop{\mathrm{arctg}}\nolimits \left( \frac{x}{\sqrt{a^{2}-x^{2}}% }\right) +C$ .
5.: $\displaystyle-\frac{1}{3}e^{3\cos x}+C$ .
6.: $\displaystyle\frac{1}{2}\left( 5+x^{3}\right) ^{\frac{2}{3}}+C$ .
7.: $\displaystyle2\sqrt{x}+\frac{\ln ^{2}\left\vert x\right\vert }{2}+C$ .
8.: $\displaystyle-\frac{1}{6}\ln \left\vert -1+2\sin x\right\vert +C$ .
9.: $\displaystyle\ln \left( \cos x\right) +\ln \left( \sin x\right) +C$ .
10.: $\displaystyle\frac{x}{2}-\frac{1}{4}\sin \left( 2x\right) +C$ .
11.: $\displaystyle\frac{1}{2}\left( x+\cos x\sin x\right) +C$ .
12.: $\displaystyle\frac{2}{3}\left( \sin x\right) ^{\frac{3}{2}}-\frac{4}{7}\left( \sin x\right) ^{\frac{7}{2}}-\frac{2}{11}\left( \sin x\right) ^{% \frac{11}{2}}+C$ .
13.: $\displaystyle e^{\mathop{\mathrm{arctg}}\nolimits x}+\mathop{\mathrm{arctg}}\nolimits x+% \frac{1}{4}{\ln }^{2}\left( 1+x^{2}\right) +C$ .
14.: $\displaystyle e^{x}\left( 2-2x+x^{2}\right) +C$ .
15.: $\displaystyle e^{x}\left( 3+x^{2}\right) +C$ .
16.: $\displaystyle-x+x\ln \left\vert x\right\vert +C$ .
17.: $\displaystyle-\frac{1+2\ln \left\vert x\right\vert }{4x^{2}}+C$ .
18.: $\displaystyle-\frac{1}{3}\sqrt{1-x^{2}}\left( 2+x^{2}\right) +C$ .
19.: $\displaystyle-\frac{1}{9}\sqrt{1-x^{2}}\left( 2+x^{2}\right) +\frac{1}{3}\arccos x+C$ .
20.: $\displaystyle\frac{x}{2a^{2}\left( a^{2}+x^{2}\right) }+\frac{ \mathop{\mathrm{arctg}}\nolimits \left( \frac{x}{a}\right) }{2a^{3}}+C$ .
21.: $\displaystyle\frac{1}{2}x\left( \cos \left( \ln \left\vert x\right\vert \right) +\sin \left( \ln \left\vert x\right\vert \right) \right) +C$ .
22.: $\displaystyle \mathop{\mathrm{arctg}}\nolimits \left( 2x+3\right) +C$ .
23.: $\displaystyle\frac{1}{54}\mathop{\mathrm{arctg}}\nolimits \frac{x+1}{3}+\frac{1}{18}\frac{x+1}{x^{2}+2x+10}% +C$ .
24.: $\displaystyle x+\frac{1}{3}\mathop{\mathrm{arctg}}\nolimits x-\frac{8}{3}\mathop{\mathrm{arctg}}\nolimits \frac{1}{2}x+C$ .
25.: $\displaystyle\frac{2}{9}\ln \left\vert x-1\right\vert -\frac{2}{9}\ln \left\vert x+2\right\vert -\frac{1}{3x-3}+C$ .
26.: $-\displaystyle\frac{4}{7}x+\frac{1}{49}\ln \left\vert x-\frac{5}{7}\right\vert +C$ .
27.: $\displaystyle 3\ln \left\vert x\right\vert -\ln \left\vert x-1\right\vert -\frac{2}{% x-1}+C$ .
28.: $\displaystyle\ln \left\vert x\right\vert -2\ln \left\vert x+1\right\vert +\ln ... ...t +\frac{2}{\sqrt{3}}\mathop{\mathrm{arctg}}\nolimits \frac{2x-1}{\sqrt{% 3}}+C$ .
29.: $\displaystyle\frac{3}{8}x-\frac{1}{4}\sin 2x+\frac{1}{32}\sin 4x+C$ .
30.: $\displaystyle-\frac{1}{3}{\mathop{\mathrm{ctg}}\nolimits } ^{3}x-3\mathop{\mat... ...op{\mathrm{tg}}\nolimits x+\frac{1}{3}{\mathop{\mathrm{tg}}\nolimits } ^{3}x+C$ .
31.: $\displaystyle\frac{1}{\sqrt{2}}\ln \left\vert \frac{1+\sqrt{2}\cos x}{1-\sqrt{... ...right\vert -\frac{1}{2}\ln \left\vert \frac{1+\cos x}{1-\cos x}% \right\vert +C$ .
32.: $\displaystyle\frac{1}{11}\sin ^{11}x-\frac{1}{13}\sin ^{13}x+C$ .
33.: $\displaystyle -\mathop{\mathrm{ctg}}\nolimits x+2\mathop{\mathrm{tg}}\nolimits x+\frac{1}{3}{\mathop{\mathrm{tg}}\nolimits } ^{3}x+C$ .
34.: $\displaystyle\frac{1}{4}\cos 2x-\frac{1}{16}\cos 8x+C$ .
35.: $\displaystyle -8\mathop{\mathrm{ctg}}\nolimits 2x-\frac{8}{3}{\mathop{\mathrm{ctg}}\nolimits } ^{3}2x+C$ .
36.: $\displaystyle\frac{1}{\sqrt{2}}\mathop{\mathrm{arctg}}\nolimits \frac{\mathop{\mathrm{tg}}\nolimits 2x}{\sqrt{2}}+C$ .
37.: $\displaystyle\frac{1}{\sqrt{2}}\ln \left\vert \frac{\cos 4x+7+4\sqrt{2}}{\cos 4x+7-4\sqrt{2}}\right\vert +C$ .
38.: $\displaystyle\frac{1}{3}\ln \left\vert \mathop{\mathrm{tg}}\nolimits \frac{x}{... ...5}{3}\ln \left\vert \mathop{\mathrm{tg}}\nolimits \frac{x}{2}% -3\right\vert +C$ .
39.: $\displaystyle\ln \left\vert \sin x\right\vert -\sin x+C$ .
40.: $\displaystyle\frac{1}{4}\ln \left\vert \sin x+\cos x\right\vert -\frac{1}{4}\cos x\left( \sin x+\cos x\right) +C$ .
41.: $\displaystyle\frac{1}{16}x-\frac{1}{192}\sin 6x-\frac{1}{192}\sin 12x+\frac{1}{576}% \sin 18x+C$ .
42.: $\displaystyle \frac{3}{2}\sqrt[3] {x^2}+\mathop{\mathrm{arctg}}\nolimits \sqrt[6] x +C$ .
43.: $\displaystyle 2\sqrt x-2\ln \vert 1+\sqrt x\vert+C$ .
44.: $\displaystyle -\frac{6}{7}(x+1)^{\frac{7}{6}}+\frac{6}{5}(x+1)^{\frac{5}{6}}+ ... ...1)^{\frac{2}{3}}-2(x+1)^{\frac{1}{2}}-3(x+1)^{\frac{1}{3}}+6(x+1)^{\frac{1}{6}}$ $+3\ln \vert\sqrt[3] {x+1}+1\vert-6\mathop{\mathrm{arctg}}\nolimits \sqrt[6] {x+1}+C$ .
45.: $\displaystyle \frac{10}{9}\cdot \frac{x-2}{\sqrt {7x-10-x^2}}-\frac{4}{9}\cdot \frac{\sqrt {7x-10-x^2}}{x-2}+C$ .
46.: $\displaystyle 2\ln \vert\sqrt {x^2-x+1}-x\vert-\frac{3}{2}\ln \vert 2\sqrt {x^2-x+1}-2x+1\vert+\frac{3}{2}\cdot \frac{1}{2\sqrt {x^2-x+1}-2x+1}$ .
47.: $\displaystyle -\frac{2}{1+\sqrt {\frac{1-x}{1+x}}}-2\mathop{\mathrm{arctg}}\nolimits \sqrt {\frac{1-x}{1+x}}+C$ .
48.: $\displaystyle \left(\frac{1}{2}x-\frac{1}{4}\right)\sqrt {4x^2-4x+3}+\frac{1}{2}\ln \vert 2x-1+\sqrt {4x^2-4x+3}\vert+C$ .
49.: $\displaystyle (-\frac{1}{3}x^2-\frac{5}{6}x-\frac{19}{6}\big(g)\sqrt{1+2x-x^2}+4\arcsin\frac{x-1}{\sqrt{2}}+C$ .
50.: $\displaystyle \left(\frac{1}{3}x^2+\frac{1}{6}x+\frac{7}{6}\right)\sqrt {x^2+2x+2}+\frac{5}{2}\ln \vert x+1+\sqrt {x^2+2x+2}\vert+C$ .
51.: $\displaystyle -\frac{1}{2(\sqrt[4] x+1)^8}+\frac{4}{9(\sqrt[4] x+1)^9}+C$ .
52.: $\displaystyle 2(1+\sqrt[3] x)^{\frac{3}{2}}+C$ .
53.: $\displaystyle I_n=-\frac{1}{n} \cos x \sin ^{n-1} x+\frac{n-1}{n}I_{n-2}$ , $n\geq 2$ ,
$\displaystyle I_4=-\frac{1}{4} \sin ^3 x \cos x-\frac{3}{8} \sin x\cos x+\frac{3}{8}x+C$ .
54.: $\displaystyle I_n=x\ln ^n x-nI_{n-1}$ .
55.: $\displaystyle I_n=\frac{1}{a}x^ne^{ax}-\frac{n}{a}I_{n-1}$ .
56.: $\displaystyle \ln (1+x)=\sum\limits_{n=0}^{\infty} (-1)^n\frac{x^{n+1}}{n+1}$ , $x\in \left<-1, 1\right]$ .
57.: $\displaystyle \sum\limits_{n=0}^{\infty} (-1)^n\frac{x^{n+1}}{(n+1)^2}$ , $x\in \left<-1, 1\right]$ .

Zadaci za vježbu NEODREĐENI INTEGRAL ODREĐENI INTEGRAL