☰ matematika2

Integriranje razvojem u red NEODREĐENI INTEGRAL Rješenja zadataka za vježbu

Zadaci za vježbu

Izračunajte integrale:

1.: $\displaystyle\int \frac{x^2+5x-1}{\sqrt{x}} dx$
2.: $\displaystyle\int \frac{2^{x+1}-5^{x-1}}{10^x} dx$
3.: $\displaystyle\int \frac{ dx}{x^2+a^2}$
4.: $\displaystyle\int \frac{ dx}{\sqrt{a^2-x^2}}$
5.: $\displaystyle\int e^{3\cos x}\sin x dx$
6.: $\displaystyle\int \frac{x^2}{\sqrt[3]{5+x^3}} dx$
7.: $\displaystyle\int \frac{\sqrt{x}+\ln x}{x} dx$
8.: $\displaystyle\int \frac{e^{2x}}{1-3e^{2x}} dx$
9.: $\displaystyle\int \frac{\cos 2x}{\sin x\cos x} dx$
10.: $\displaystyle\int \sin^2 x dx$
11.: $\displaystyle\int \cos^2 x dx$
12.: $\displaystyle\int \cos^5 x\sqrt{\sin x} dx$
13.: $\displaystyle\int \frac{e^{\mathop{\mathrm{arctg}}\nolimits {x}}+x\ln(1+x^2)+1}{1+x^2} dx$
14.: $\displaystyle\int x^2e^x dx$
15.: $\displaystyle\int (x^2+2x+3)e^x dx$
16.: $\displaystyle\int \ln x dx$
17.: $\displaystyle\int \frac{\ln x}{x^3} dx$
18.: $\displaystyle\int \frac{x^3}{\sqrt{1-x^2}} dx$
19.: $\displaystyle\int x^2\arccos x dx$
20.: $\displaystyle\int \frac{ dx}{(x^2+a^2)^2}$
21.: $\displaystyle\int \cos{(\ln x)} dx$
22.: $\displaystyle\int \frac{ dx}{2x^{2}+6x+5}$
23.: $\displaystyle\int \frac{ dx}{\left( x^{2}+2x+10\right) ^{2}}$
24.: $\displaystyle\int \frac{x^{4} dx}{x^{4}+5x^{2}+4}$
25.: $\displaystyle\int \frac{x dx}{x^{3}-3x+2}$
26.: $\displaystyle\int \frac{4x-3}{5-7x} dx$
27.: $\displaystyle\int \frac{2x^{2}-3x+3}{x^{3}-2x^{2}+x} dx$
28.: $\displaystyle\int \frac{x^{3}+4x^{2}-2x+1}{x^{4}+x} dx$
29.: $\displaystyle\int \sin ^{4}x dx$
30.: $\displaystyle\int \frac{ dx}{\sin ^{4}x\cos ^{4}x}$
31.: $\displaystyle\int \frac{ dx}{\sin x\left( 2\cos ^{2}x-1\right) }$
32.: $\displaystyle\int \sin ^{10}x\cos ^{3}x dx$
33.: $\displaystyle\int\displaystyle \frac{ dx}{\sin ^{2}x\cos ^{4}x}$
34.: $\displaystyle\int \sin 3x\cos 5x dx$
35.: $\displaystyle\int \frac{ dx}{\sin ^{4}x\cos ^{4}x}$
36.: $\displaystyle\int \frac{ dx}{\sin ^{4}x+\cos ^{4}x}$
37.: $\displaystyle\int \frac{\sin 4x}{\sin ^{8}x+\cos ^{8}x} dx$
38.: $\displaystyle\int \frac{ dx}{\sin x\left( 2+\cos x-2\sin x\right) }$
39.: $\displaystyle\int \frac{\cos ^{3}x}{\sin ^{2}x+\sin x} dx$
40.: $\displaystyle\int \frac{\sin ^{2}x\cos x}{\sin x+\cos x} dx$
41.: $\displaystyle\int \sin ^{4}3x\cos ^{2}3x dx$
42.: $\displaystyle \int \frac{x+\sqrt[3] {x^2}+\sqrt[6] x}{x(1+\sqrt[3] x)} dx$ .
43.: $\displaystyle \int \frac{ dx}{1+\sqrt x}$
44.: $\displaystyle \int \frac{1-\sqrt {x+1}}{1+\sqrt[3] {x+1}} dx$ .
45.: $\displaystyle \int \frac{x dx}{(\sqrt {7x-10-x^2})^3}$ .
46.: $\displaystyle \int \frac{ dx}{x-\sqrt {x^2-x+1}}$ .
47.: $\displaystyle \int \frac{ dx}{\sqrt {1-x^2} +1}$ .
48.: $\displaystyle \int \sqrt {4x^2-4x+3} dx$ .
49.: $\displaystyle \int \frac{x^3}{\sqrt {1+2x-x^2}} dx$ .
50.: $\displaystyle \int \frac{x^3+2x^2+3x+4}{\sqrt {x^2+2x+2}} dx$ .
51.: $\displaystyle \int \frac{ dx}{\sqrt x(\sqrt[4] x+1)^{10}}$ .
52.: $\displaystyle \int \frac{\sqrt {1+\sqrt[3] x}}{\sqrt[3] {x^2}} dx$ .
53.: Odredite rekurzivnu formulu za integral $I_n=\int \sin ^n x dx$ . Koristeći se dobivenim rezultatom izračunajte vrijednost integrala $\int \sin^4 x dx$ .
54.: Odredite rekurzivnu formulu za integral $I_n=\int (\ln x)^n dx$ .
55.: Odredite rekurzivnu formulu za integral $I_n=\int x^n e^{ax} dx$ .
56.: Razvijte u red potencija funkciju $\displaystyle \ln (1+x)$ pomoću $\displaystyle \int \frac{1}{1+x} dx$ .
57.: Odredite $\displaystyle \int \frac{\ln (1+x)}{x} dx$ razvojem podintegralne funkcije u red potencija.

Integriranje razvojem u red NEODREĐENI INTEGRAL Rješenja zadataka za vježbu