Integrale & Derivate

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43 Terms

1
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∫dx\int dx

x+Cx + C

2
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∫x dx\int x\,dx

x22+C\frac{x^2}{2} + C

3
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∫xn dx\int x^n\,dx

xn+1n+1+C\frac{x^{n+1}}{n+1} + C

4
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∫x dx\int \sqrt{x}\,dx

23xx+C\frac{2}{3}x\sqrt{x} + C

5
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∫ex dx\int e^x\,dx

ex+Ce^x + C

6
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∫ax dx\int a^x\,dx

axln⁡(a)+C\frac{a^x}{\ln(a)} + C

7
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∫1x dx\int \frac{1}{x}\,dx

ln⁡(∣x∣)+C\ln(|x|) + C

8
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∫1x2−a2 dx\int \frac{1}{x^2 - a^2}\,dx

12aln⁡∣x−ax+a∣+C\frac{1}{2a}\ln\left|\frac{x - a}{x + a}\right| + C

9
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∫1x2+1 dx\int \frac{1}{x^2 + 1}\,dx

arctg⁡(x)+C\operatorname{arctg}(x) + C

10
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∫1x2+a2 dx\int \frac{1}{x^2 + a^2}\,dx

1aarctg⁡(xa)+C\frac{1}{a}\operatorname{arctg}\left(\frac{x}{a}\right) + C

11
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∫1x2−a2 dx\int \frac{1}{\sqrt{x^2 - a^2}}\,dx

ln⁡∣x+x2−a2∣+C\ln\left|x + \sqrt{x^2 - a^2}\right| + C

12
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∫1x2+a2 dx\int \frac{1}{\sqrt{x^2 + a^2}}\,dx

ln⁡(x+x2+a2)+C\ln\left(x + \sqrt{x^2 + a^2}\right) + C

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∫11−x2 dx\int \frac{1}{\sqrt{1 - x^2}}\,dx

arcsin⁡(x)+C\arcsin(x) + C

14
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∫1a2−x2 dx\int \frac{1}{\sqrt{a^2 - x^2}}\,dx

arcsin⁡(xa)+C\arcsin\left(\frac{x}{a}\right) + C

15
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∫sin⁡(x) dx\int \sin(x)\,dx

−cos⁡(x)+C-\cos(x) + C

16
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∫cos⁡(x) dx\int \cos(x)\,dx

sin⁡(x)+C\sin(x) + C

17
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∫tg⁡(x) dx\int \operatorname{tg}(x)\,dx

−ln⁡(∣cos⁡(x)∣)+C-\ln(|\cos(x)|) + C

18
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∫ctg⁡(x) dx\int \operatorname{ctg}(x)\,dx

ln⁡(∣sin⁡(x)∣)+C\ln(|\sin(x)|) + C

19
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∫1cos⁡2(x) dx\int \frac{1}{\cos^2(x)}\,dx

tg⁡(x)+C\operatorname{tg}(x) + C

20
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∫1sin⁡2(x) dx\int \frac{1}{\sin^2(x)}\,dx

−ctg⁡(x)+C-\operatorname{ctg}(x) + C

21
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∫xx2−a2 dx\int \frac{x}{\sqrt{x^2 - a^2}}\,dx

x2−a2+C\sqrt{x^2 - a^2} + C

22
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∫xx2+a2 dx\int \frac{x}{\sqrt{x^2 + a^2}}\,dx

x2+a2+C\sqrt{x^2 + a^2} + C

23
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∫xa2−x2 dx\int \frac{x}{\sqrt{a^2 - x^2}}\,dx

−a2−x2+C-\sqrt{a^2 - x^2} + C

24
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(c)′(c)'

00

25
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(x)′(x)'

11

26
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(xn)′(x^n)'

nxn−1n x^{n-1}

27
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(x)′(\sqrt{x})'

12x\frac{1}{2\sqrt{x}}

28
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(1x)′\left(\frac{1}{x}\right)'

−1x2-\frac{1}{x^2}

29
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(ex)′(e^x)'

exe^x

30
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(ax)′(a^x)'

axln⁡(a)a^x \ln(a)

31
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(ln⁡(x))′(\ln(x))'

1x\frac{1}{x}

32
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(log⁡a(x))′(\log_a(x))'

1xln⁡(a)\frac{1}{x \ln(a)}

33
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(arctg⁡(x))′(\operatorname{arctg}(x))'

1x2+1\frac{1}{x^2 + 1}

34
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(arcctg⁡(x))′(\operatorname{arcctg}(x))'

−1x2+1-\frac{1}{x^2 + 1}

35
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(arcsin⁡(x))′(\arcsin(x))'

11−x2\frac{1}{\sqrt{1 - x^2}}

36
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(arccos⁡(x))′(\arccos(x))'

−11−x2-\frac{1}{\sqrt{1 - x^2}}

37
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(sin⁡(x))′(\sin(x))'

cos⁡(x)\cos(x)

38
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(cos⁡(x))′(\cos(x))'

−sin⁡(x)-\sin(x)

39
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(tg⁡(x))′(\operatorname{tg}(x))'

1cos⁡2(x)\frac{1}{\cos^2(x)}

40
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(ctg⁡(x))′(\operatorname{ctg}(x))'

−1sin⁡2(x)-\frac{1}{\sin^2(x)}

41
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(x2−a2)′(\sqrt{x^2 - a^2})'

xx2−a2\frac{x}{\sqrt{x^2 - a^2}}

42
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(x2+a2)′(\sqrt{x^2 + a^2})'

xx2+a2\frac{x}{\sqrt{x^2 + a^2}}

43
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(a2−x2)′(\sqrt{a^2 - x^2})'

−xa2−x2-\frac{x}{\sqrt{a^2 - x^2}}