TABLICE CAŁEK - WZORY
Wzory podstawowe
1. \(\int 0 d x=C\)
2. \(\int d x=x+C\)
3. \(\int x d x=\frac{1}{2} x^{2}+C\)
4. \(\int x^{n} d x=\frac{1}{n+1} x^{n+1}+C\), dla \(n \neq-1\)
5. \(\int \frac{1}{x} d x=\ln |x|+C\)
6. \(\int \frac{f^{\prime}(x)}{f(x)} d x=\ln |f(x)|+C\)
7. \(\int \frac{1}{x^{2}} d x=-\frac{1}{x}+C\)
8. \(\int \sqrt{x} d x=\frac{2}{3} x \sqrt{x}\)
9. \(\int \frac{1}{\sqrt{x}} d x=2 \sqrt{x}+C\)
10. \(\int \frac{f^{\prime}(x)}{\sqrt{f(x)}} d x=2 \sqrt{f(x)}+C\)
11. \(\int \frac{d x}{\sqrt{1-x^{2}}}=\arcsin x+C\)
12. \(\int \sin x d x=-\cos x+C\)
13. \(\int{ } \sinh x d x=-{ } \cosh x+C\)
14. \(\int \cos x d x=\sin x+C\)
15. \(\int \cosh x d x=\sinh x+C\)
16. \(\int \frac{1}{\sin ^{2} x} d x=-{ } \cot x+C\)
17. \(\int \frac{1}{\sinh ^{2} x} d x=-{ } \operatorname{coth} x+C\)
18. \(\int \frac{1}{\cos ^{2} x} d x=\tan x+C\)
19. \(\int \frac{1}{\cosh ^{2} x} d x={ } \tanh x+C\)
20. \(\int e^{x} d x=e^{x}+C\)
21. \(\int \ln x d x=x \ln x-x+C\)
22. \(\int \arctan x d x=x \arctan x-\ln \sqrt{x^{2}+1}\)
21. \(\int m^{x} d x=\frac{m^{x}}{\ln m}+C\), dla \(m>0\) i \(m \neq 1\)
gdzie
\(\sinh x=\frac{e^{x}-e^{-x}}{2}\), jest to sinus hiperboliczy
\(\cosh x=\frac{e^{x}+e^{-x}}{2}\), jest to cosinus hiperboliczy
\(\cot x\) oznacza cotangens
\(\coth x=\frac{\cosh x}{\sinh x}\), jest to cotangens hiperboliczy
\(\tanh x=\frac{\sinh x}{\cosh x}\), jest to tangens hiperboliczy
Całkowanie funkcji wielomianowych
1. \(\int 0 d x=C\)
2. \(\int d x=x+C\)
3. \(\int x d x=\frac{1}{2} x^{2}+C\)
4. \(\int(a x+b) d x=\frac{a}{2} x^{2}+b x+C\)
5. \(\int x^{n} d x=\frac{1}{n+1} x^{n+1}+C\), dla \(n \neq-1\)
6. \(\int(a x+b)^{n} d x=\frac{1}{a(n+1)}(a x+b)^{n+1}+C\), dla \(a \neq 0\) i \(n \neq-1\)
7. \(\int\left(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}\right) d x=\frac{a_{n}}{n+1} x^{n+1}+\frac{a_{n-1}}{n} x^{n}+\) \(\ldots+\frac{a_{1}}{2} x^{2}+a_{0} x+C\)
Całkowanie funkcji wymiernych
1. \(\int \frac{1}{x} d x=\ln |x|+C\)
2. \(\int \frac{1}{x^{2}} d x=-\frac{1}{x}+C\)
3. \(\int \frac{d x}{1+x^{2}}=\arctan x+C\)
4. \(\int \frac{d x}{\left(1+x^{2}\right)^{n}}=\frac{x}{2(n-1)\left(1+x^{2}\right)^{n-1}}+\frac{2 n-3}{2 n-2} \int \frac{d x}{\left(1+x^{2}\right)^{n-1}}\), dla \(n \neq 1\)
5. \(\int \frac{d x}{1+(a x+b)^{2}}=\frac{1}{a} \arctan (a x+b)+C\), dla \(a \neq 0\)
6. \(\int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \arctan \frac{x}{a}+C\), dla \(a \neq 0\)
7. \(\int \frac{d x}{b+(x-a)^{2}}=\frac{1}{\sqrt{b}} \arctan \frac{x-a}{\sqrt{b}}+C\), dla \(b>0\)
8. \(\int \frac{d x}{a^{2}-x^{2}}=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+C\), dla \(a>0\) i \(|x| \neq 0\)
9. \(\int \frac{1}{a x+b} d x=\frac{1}{a} \ln |a x+b|+C\), dla \(a \neq 0\)
10. \(\int \frac{1}{(a x+b)^{2}} d x=-\frac{1}{a(a x+b)}+C\)
11. \(\int \frac{1}{(a x+b)^{n}}=\frac{1}{a(1-n)(a x+b)^{n-1}}+C\), dla \(n \neq 1\)
12. \(\int \frac{A x+B}{a x+b} d x=\frac{A}{a} x+\frac{a B-A b}{a^{2}} \ln |a x+b|+C\), dla \(a \neq 0\)
13. \(\int \frac{d x}{a x^{2}+b x+c}=\frac{1}{a \sqrt{\frac{-\Delta}{4 a^{2}}}} \arctan \frac{x+\frac{b}{2 a}}{\sqrt{\frac{-\Delta}{4 a^{2}}}}+C\), dla \(a \neq 0\) oraz \(\Delta<0\)
14. \(\int \frac{d x}{a x^{2}+b x+c}=\frac{1}{\sqrt{\Delta}} \ln \left|\frac{x+\frac{b-\sqrt{\Delta}}{2 a}}{x+\frac{b+\sqrt{\Delta}}{2 a}}\right|+C\), dla \(a \neq 0\) oraz \(\Delta>0\)
15. \(\int \frac{d x}{a x^{2}+b x+c}=-\frac{1}{a x+\frac{b}{2}}+C\), dla \(a \neq 0\) oraz \(\Delta=0\)
16. \(\int \frac{d x}{b+x^{2}}=\frac{1}{\sqrt{b}} \arctan \frac{x}{\sqrt{b}}+C\), dla \(b>0\)
17. \(\int \frac{A x+B}{a x^{2}+b x+c} d x=\frac{A}{2 a} \ln \left|a x^{2}+b x+c\right|+\frac{2 a B-A b}{a \sqrt{-\Delta}} \arctan \frac{x+\frac{b}{2 a}}{\sqrt{\frac{-\Delta}{4 a^{2}}}}+C\), dla \(a \neq 0\) oraz \(\Delta<0\)
18. \(\int \frac{A x+B}{a x^{2}+b x+c} d x=\frac{A}{2 a} \ln \left|a x^{2}+b x+c\right|+\frac{2 a B-A b}{2 a \sqrt{\Delta}} \ln \left|\frac{x+\frac{b-\sqrt{\Delta}}{2 a}}{x+\frac{b+\sqrt{\Delta}}{2 a}}\right|+C\), dla \(a \neq 0\) oraz \(\Delta>0\)
19. \(\int \frac{A x+B}{a x^{2}+b x+c} d x=\frac{A}{2 a} \ln \left|a x^{2}+b x+c\right|+\frac{2 a B-A b}{2 a}\left(-\frac{1}{a x+\frac{b}{2}}\right)+C\), dla \(a \neq 0 \operatorname{oraz} \Delta=0\)
20. \(\int \frac{A x+B}{\left(a x^{2}+b x+c\right)^{n}} d x=\frac{A}{2 a(1-n)\left(a x^{2}+b x+c\right)^{n-1}}+\frac{2 a B-b A}{2 a^{n+1}\left(\frac{-\Delta}{4 a^{2}}\right)^{n-\frac{1}{2}}} \int \frac{d t}{\left(1+t^{2}\right)^{n}}\), dla \(a \neq 0, n \neq 1, \Delta<0\) oraz \(t=\frac{x+\frac{b}{2 a}}{\sqrt{\frac{-\Delta}{4 a^{2}}}}\)
21. \(\int \frac{A x^{2}+B x+C}{a x^{2}+b x+c} d x=\frac{A}{a} x+\frac{B-\frac{b A}{a}}{2 a} \ln \left|a x^{2}+b x+c\right|+\frac{2 a\left(C-\frac{c A}{a}\right)-\left(B-\frac{b A}{a}\right) b}{a \sqrt{-\Delta}} \arctan \frac{x+\frac{b}{2 a}}{\sqrt{\frac{-\Delta}{4 a^{2}}}}+\) \(C\), dla \(a \neq 0\) oraz \(\Delta<0\)
22. \(\int \frac{A x^{2}+B x+C}{a x^{2}+b x+c} d x=\frac{A}{a} x+\frac{B-\frac{b A}{a}}{2 a} \ln \left|a x^{2}+b x+c\right|+\frac{2 a\left(C-\frac{c A}{a}\right)-\left(B-\frac{b A}{a}\right) b}{2 a \sqrt{\Delta}} \ln \left|\frac{x+\frac{b-\sqrt{\Delta}}{2 a}}{x+\frac{b+\sqrt{\Delta}}{2 a}}\right|+\) \(C\), dla \(a \neq 0\) oraz \(\Delta>0\)
23. \(\int \frac{A x^{2}+B x+C}{a x^{2}+b x+c} d x=\frac{A}{a} x+\frac{B-\frac{b A}{a}}{2 a} \ln \left|a x^{2}+b x+c\right|+\frac{2 a\left(C-\frac{c A}{a}\right)-\left(B-\frac{b A}{a}\right) b}{2 a}\left(-\frac{1}{a x+\frac{b}{2}}\right)+\) \(C\), dla \(a \neq 0\) oraz \(\Delta=0\)
24. \(\int \frac{d x}{(x-a)(x-b)(x-c)}=\frac{1}{(a-b)(a-c)} \ln |x-a|+\frac{1}{(b-a)(b-c)} \ln |x-b|+\frac{1}{(c-a)(c-b)} \ln |x-c|+\) \(C\), dla \(a \neq b \neq c\)
25. \(\int \frac{A x+B}{(x-a)(x-b)(x-c)} d x=\frac{A a+B}{(a-b)(a-c)} \ln |x-a|+\frac{A b+B}{(b-a)(b-c)} \ln |x-b|+\) \(\frac{A c+B}{(c-a)(c-b)} \ln |x-c|+C\), dla \(a \neq b \neq c\)
Całkowanie funkcji niewymiernych
1. \(\int \sqrt{x} d x=\frac{2}{3} x \sqrt{x}\)
2. \(\int \sqrt{a x+b} d x=\frac{2}{3 a}(a x+b) \sqrt{(a x+b)}\), dla \(a \neq 0\)
3. \(\int \frac{1}{\sqrt{x}} d x=2 \sqrt{x}+C\)
4. \(\int \frac{1}{\sqrt{(a x+b)}} d x=\frac{2 \sqrt{a x+b}}{a}+C\), dla \(a \neq 0\)
5. \(\int \frac{d x}{\sqrt{1-x^{2}}}=\arcsin x+C\)
6. \(\int \frac{d x}{\sqrt{1-(a x+b)^{2}}}=\frac{1}{a} \arcsin (a x+b)+C\), dla \(a \neq 0\)
7. \(\int \frac{d x}{\sqrt{a^{2}-x^{2}}}=\arcsin \frac{x}{a}+C\), dla \(a>0\)
8. \(\int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C\), dla \(a \neq 0\)
9. \(\int \frac{d x}{\sqrt{1+x^{2}}}=\ln \left(x+\sqrt{x^{2}+1}\right)+C\)
10. \(\int \frac{d x}{\sqrt{1+(a x+b)^{2}}}=\frac{1}{a} \ln \left((a x+b)+\sqrt{(a x+b)^{2}+1}\right)+C\), dla \(a \neq 0\)
11. \(\int \frac{d x}{\sqrt{x^{2}-1}}=\ln \left|x+\sqrt{x^{2}-1}\right|+C\), dla \(|x|>1\)
12. \(\int \frac{d x}{\sqrt{(a x+b)^{2}-1}}=\frac{1}{a} \ln \left|(a x+b)+\sqrt{(a x+b)^{2}-1}\right|+C\), dla \(\mid a x+\) \(b \mid>1\) i \(a \neq 0\)
13. \(\int \frac{d x}{\sqrt{x^{2}+b x+c}}=\ln \left|x+\frac{1}{2} b+\sqrt{x^{2}+b x+c}\right|+C\), dla \({ } \Delta<0\)
14. \(\int \frac{d x}{\sqrt{a x^{2}+b x+c}}=\frac{1}{\sqrt{-a}} \arcsin \frac{\sqrt{-a} x-\frac{b}{2 \sqrt{-a}}}{\sqrt{\frac{\Delta}{-4 a}}}+C\), dla \(a<0\), oraz \(\Delta>0\)
15. \(\int \frac{d x}{0}=\frac{1}{\sqrt{a} x^{2}+b x+c} \ln \left|\sqrt{a} x+\frac{b}{2 \sqrt{a}}+\sqrt{a x^{2}+b x+c}\right|+C\), dla \(a>\) 0 i \(\Delta<0\)
16. \(\int \frac{A x+B}{\sqrt{a x^{2}+b x+c}} d x=\frac{A}{a} \sqrt{a x^{2}+b x+c}+\frac{2 a B-A b}{2 a \sqrt{a}} \ln \left|\sqrt{a} x+\frac{b}{2 \sqrt{a}}+\sqrt{a x^{2}+b x+c}\right|+\) \(C\), dla \(a>0\) i \(\Delta<0\)
17. \(\int \frac{A x+B}{\sqrt{a x^{2}+b x+c}} d x=\frac{A}{a} \sqrt{a x^{2}+b x+c}+\frac{2 a B-A b}{2 a \sqrt{-a}} \arcsin \frac{\sqrt{-a} x-\frac{b}{2 \sqrt{-a}}}{\sqrt{\frac{\Delta}{-4 a}}}+\) \(C\), dla \(a<0\), oraz \(\Delta>0\)
Całkowanie funkcji trygonometrycznych
1. \(\int \sin x d x=-\cos x+C\)
2. \(\int \sin (a x+b) d x=-\frac{1}{a} \cos (a x+b)+C\), dla \(a \neq 0\)
3. \(\int \cos x d x=\sin x+C\)
4. \(\int \cos (a x+b) d x=\frac{1}{a} \sin (a x+b)+C\), dla \(a \neq 0\)
5. \(\int \frac{1}{\sin ^{2} x} d x=-\cot x+C\)
6. \(\int \frac{1}{\sin ^{2}(a x+b)} d x=-\frac{1}{a} \cot (a x+b)+C\), dla \(a \neq 0\)
7. \(\int \frac{1}{\cos ^{2} x} d x=\tan x+C\)
8. \(\int \frac{1}{\cos ^{2}(a x+b)} d x=\frac{1}{a} \tan (a x+b)+C\), dla \(a \neq 0\)
9. \(\int \sinh x d x=-\cosh x+C\)
10. \(\int \sinh (a x+b) d x=-\frac{1}{a} \cosh (a x+b)+C\), dla \(a \neq 0\)
11. \(\int \cosh x d x=\sinh x+C\)
12. \(\int \cosh (a x+b) d x=\frac{1}{a} \sinh (a x+b)+C\), dla \(a \neq 0\)
13. \(\int \frac{1}{\cosh ^{2} x} d x=\tanh x+C\)
14. \(\int \frac{1}{\cosh ^{2}(a x+b)} d x=\frac{1}{a} \tanh (a x+b)+C\), dla \(a \neq 0\)
15. \(\int \frac{1}{\sinh ^{2} x} d x=-\operatorname{coth} x+C\)
16. \(\int \frac{1}{\sinh ^{2}(a x+b)} d x=-\frac{1}{a} \operatorname{coth}(a x+b)+C\), dla \(a \neq 0\)
Całkowanie funkcji wykładniczych
1. \(\int e^{x} d x=e^{x}+C\)
2. \(\int e^{a x+b} d x=\frac{1}{a} e^{a x+b}+C\), dla \(a \neq 0\)
3. \(\int m^{x} d x=\frac{m^{x}}{\ln m}+C\), dla \(m>0\) i \(m \neq 1\)
4. \(\int m^{a x+b} d x=\frac{m^{a x+b}}{a \ln m}+C\), dla \(d>0, m \neq 1\) i \(a \neq 0\)
Całkowanie przez cz?ści i podstawienie
1. \(\int \ln (a x+b) d x=\frac{1}{a}[(a x+b) \ln (a x+b)-(a x+b)]+C\), dla \(a \neq 0\)
2. \(\int x^{n} \ln x d x=\frac{1}{n+1} x^{n+1} \ln x-\frac{1}{(n+1)^{2}} x^{n+1}+C\)
3. \(\int \arctan (a x+b) d x=\frac{1}{a}\left[(a x+b) \arctan (a x+b)-\ln \sqrt{(a x+b)^{2}+1}\right]+C\)
