Bài 1: NGUYÊN HÀM

1. $int{0dx=C}$       2. $int{dx=x+C}$           

3. $int{{{x}^{alpha }}dx=frac{1}{alpha +1}{{x}^{alpha +1}}+C}left$alphane-1right$$

16. $int{{{left$ax+bright$}^{alpha }}operatorname{dx}}=frac{1}{a}frac{{{left$ax+bright$}^{alpha +1}}}{alpha +1}+c,,alpha ne -1$

4. $int{frac{1}{{{x}^{2}}}dx=-frac{1}{x}+C}$

17. $int{xdx=frac{{{x}^{2}}}{2}+C}$

5. $int{frac{1}{x}dx=ln left| x right|+C}$

18. $int{frac{operatorname{dx}}{ax+b}}=frac{1}{a}ln left| ax+b right|+c$

6. $int{{{e}^{x}}dx={{e}^{x}}+C}$

19. $int{{{e}^{ax+b}}dx=frac{1}{a}{{e}^{ax+b}}+C}$

7. $int{{{a}^{x}}dx=frac{{{a}^{x}}}{ln a}+C}$

20. $int{{{a}^{kx+b}}dx=frac{1}{k}frac{{{a}^{kx+b}}}{ln a}+C}$

8. $int{cos xdx=sin x+C}$

21. $int{cos left$ax+bright$dx=frac{1}{a}sin left$ax+bright$+C}$

9. $int{sinxdx=-cooperatorname{s}x+C}$

22. $int{sin left$ax+bright$dx=-frac{1}{a}cos left$ax+bright$+C}$

10. $int{tan x.dx,=-ln |cos x|+C}$

23.$int{tan left$ax+bright$operatorname{dx}}=-frac{1}{a}ln left| cos left$ax+bright$ right|+C$

11.$int{cot x.dx,=ln |sin x|+C}$

24.$int{cot left$ax+bright$operatorname{dx}}=frac{1}{a}ln left| sin left$ax+bright$ right|+C$

12. $int{frac{1}{{{cos }^{2}}x}dx=tan x+C}$

25. $int{frac{1}{{{cos }^{2}}left$ax+bright$}dx=frac{1}{a}tan left$ax+bright$+C}$

13.$int{frac{1}{{{sin }^{2}}x}dx=-cot x+C}$

26. $int{frac{1}{{{sin }^{2}}left$ax+bright$}dx=-frac{1}{a}cot left$ax+bright$+C}$

14.$int{left$1+{tan}^{2}xright$dx=tan x+C}$

27. $int{left$1+{tan}^{2}left(ax+bright$ right)dx=frac{1}{a}tan left$ax+bright$+C}$

15. $int{left$1+{cot}^{2}xright$dx=-cooperatorname{t}x+C}$

28.$int{left$1+{cot}^{2}left(ax+bright$ right)dx=-frac{1}{a}cooperatorname{t}left$ax+bright$+C}$

$int{frac{operatorname{dx}}{{{a}^{2}}+{{x}^{2}}}=frac{1}{a}operatorname{arctg}frac{x}{a}}+C$

$int{arcsin frac{x}{a}operatorname{dx}}=xarcsin frac{x}{a}+sqrt{{{a}^{2}}-{{x}^{2}}}+C$

$int{frac{operatorname{dx}}{{{a}^{2}}-{{x}^{2}}}=frac{1}{2a}ln left| frac{a+x}{a-x} right|}+C$

$int{arccos frac{x}{a}operatorname{dx}}=xarccos frac{x}{a}-sqrt{{{a}^{2}}-{{x}^{2}}}+C$

$int{frac{operatorname{dx}}{sqrt{{{x}^{2}}+{{a}^{2}}}}}=ln left$x+sqrt{x}^2+a^{2}right$+C$

$int{arctan frac{x}{a}operatorname{dx}}=xarctan frac{x}{a}-frac{a}{2}ln left$a^2+x^{2}right$+C$

$int{frac{operatorname{dx}}{sqrt{{{a}^{2}}-{{x}^{2}}}}}=arcsin frac{x}{left| a right|}+C$

$int{operatorname{arc}cot frac{x}{a}operatorname{dx}}=xoperatorname{arc}cot frac{x}{a}+frac{a}{2}ln left$a^2+x^{2}right$+C$

$int{frac{operatorname{dx}}{xsqrt{{{x}^{2}}-{{a}^{2}}}}}=frac{1}{a}arccos left| frac{x}{a} right|+C$

$int{frac{operatorname{dx}}{xsqrt{{{x}^{2}}+{{a}^{2}}}}}=-frac{1}{a}ln left| frac{a+sqrt{{{x}^{2}}+{{a}^{2}}}}{x} right|+C$

$int{frac{operatorname{dx}}{sin left$ax+bright$}}=frac{1}{a}ln left| tan frac{ax+b}{2} right|+C$

$int{ln left$ax+bright$operatorname{dx}}=left$x+fracbaright$ln left$ax+bright$-x+C$

$int{{{e}^{ax}}cos bxoperatorname{dx}}=frac{{{e}^{ax}}left$acosbx+bsinbxright$}{{{a}^{2}}+{{b}^{2}}}+C$

$int{sqrt{{{a}^{2}}-{{x}^{2}}}operatorname{dx}}=frac{xsqrt{{{a}^{2}}-{{x}^{2}}}}{2}+frac{{{a}^{2}}}{2}arcsin frac{x}{a}+C$

$int{{{e}^{ax}}sin bxoperatorname{dx}}=frac{{{e}^{ax}}left$asinbx-bcosbxright$}{{{a}^{2}}+{{b}^{2}}}+C$