Stammfunktionen finden

Note 1: Strategie

Um die Stammfunktion FF einer Funktion ff zu finden, schreibe es so hin:

F=?↓′f\begin{array}{c} F=?\\ \downarrow^\prime\\ f \end{array}

und probiere ein bisschen aus, was FF sein könnte, indem du die Ableitungsreglen durchgehst.

Exercise 1

Finde die Stammfunktion von ff.

  1. f(x)=5x4−3x2+2f(x) = 5x^4 - 3x^2 + 2

  2. f(x)=x7+12x3f(x) = x^7 + \frac{1}{2}x^3

  3. f(x)=sin⁡(x)+cos⁡(x)f(x) = \sin(x) + \cos(x)

  4. f(x)=4exf(x) = 4e^x

  5. f(x)=1x(x>0)f(x) = \frac{1}{x} \quad (x > 0)

  6. f(x)=2⋅cos⁡(x)−3⋅sin⁡(x)f(x) = 2 \cdot \cos(x) - 3 \cdot \sin(x)

  7. f(x)=ex+xf(x) = e^x + x

  8. f(x)=3x2f(x) = \frac{3}{x^2}

  9. f(x)=1x5f(x) = \frac{1}{x^5}

  10. f(x)=4x3+2xf(x) = \frac{4}{x^3} + 2x

  11. f(x)=xf(x) = \sqrt{x}

  12. f(x)=x3f(x) = \sqrt[3]{x}

  13. f(x)=2⋅x4f(x) = 2 \cdot \sqrt[4]{x}

  14. f(x)=1xf(x) = \frac{1}{\sqrt{x}}

  15. f(x)=x25f(x) = \sqrt[5]{x^2}

  16. f(x)=2x4−xf(x) = \frac{2}{x^4} - \sqrt{x}

  17. f(x)=5⋅cos⁡(x)+exf(x) = 5 \cdot \cos(x) + e^x

  18. f(x)=13x2f(x) = \frac{1}{3x^2}

  19. f(x)=10x9−1x2f(x) = 10x^9 - \frac{1}{x^2}

  20. f(x)=x23+5x6f(x) = \sqrt[3]{x^2} + \frac{5}{x^6}

Solution
  1. F(x)=x5−x3+2xF(x) = x^5 - x^3 + 2x
  2. F(x)=18x8+18x4F(x) = \frac{1}{8}x^8 + \frac{1}{8}x^4
  3. F(x)=−cos⁡(x)+sin⁡(x)F(x) = -\cos(x) + \sin(x)
  4. F(x)=4exF(x) = 4e^x
  5. F(x)=ln⁡(x)F(x) = \ln(x)
  6. F(x)=2sin⁡(x)+3cos⁡(x)F(x) = 2\sin(x) + 3\cos(x)
  7. F(x)=ex+12x2F(x) = e^x + \frac{1}{2}x^2
  8. f(x)=3x−2⇒F(x)=−3x−1=−3xf(x) = 3x^{-2} \Rightarrow F(x) = -3x^{-1} = -\frac{3}{x}
  9. f(x)=x−5⇒F(x)=−14x−4=−14x4f(x) = x^{-5} \Rightarrow F(x) = -\frac{1}{4}x^{-4} = -\frac{1}{4x^4}
  10. f(x)=4x−3+2x⇒F(x)=−2x−2+x2=−2x2+x2f(x) = 4x^{-3} + 2x \Rightarrow F(x) = -2x^{-2} + x^2 = -\frac{2}{x^2} + x^2
  11. f(x)=x12⇒F(x)=23x32=23x3f(x) = x^{\frac{1}{2}} \Rightarrow F(x) = \frac{2}{3}x^{\frac{3}{2}} = \frac{2}{3}\sqrt{x^3}
  12. f(x)=x13⇒F(x)=34x43=34x43f(x) = x^{\frac{1}{3}} \Rightarrow F(x) = \frac{3}{4}x^{\frac{4}{3}} = \frac{3}{4}\sqrt[3]{x^4}
  13. f(x)=2x14⇒F(x)=2⋅45x54=85x54f(x) = 2x^{\frac{1}{4}} \Rightarrow F(x) = 2 \cdot \frac{4}{5}x^{\frac{5}{4}} = \frac{8}{5}\sqrt[4]{x^5}
  14. f(x)=x−12⇒F(x)=2x12=2xf(x) = x^{-\frac{1}{2}} \Rightarrow F(x) = 2x^{\frac{1}{2}} = 2\sqrt{x}
  15. f(x)=x25⇒F(x)=57x75=57x75f(x) = x^{\frac{2}{5}} \Rightarrow F(x) = \frac{5}{7}x^{\frac{7}{5}} = \frac{5}{7}\sqrt[5]{x^7}
  16. f(x)=2x−4−x12⇒F(x)=−23x−3−23x32=−23x3−23x3f(x) = 2x^{-4} - x^{\frac{1}{2}} \Rightarrow F(x) = -\frac{2}{3}x^{-3} - \frac{2}{3}x^{\frac{3}{2}} = -\frac{2}{3x^3} - \frac{2}{3}\sqrt{x^3}
  17. F(x)=5sin⁡(x)+exF(x) = 5\sin(x) + e^x
  18. f(x)=13x−2⇒F(x)=−13x−1=−13xf(x) = \frac{1}{3}x^{-2} \Rightarrow F(x) = -\frac{1}{3}x^{-1} = -\frac{1}{3x}
  19. f(x)=10x9−x−2⇒F(x)=x10+x−1=x10+1xf(x) = 10x^9 - x^{-2} \Rightarrow F(x) = x^{10} + x^{-1} = x^{10} + \frac{1}{x}
  20. f(x)=x23+5x−6⇒F(x)=35x53−x−5=35x53−1x5f(x) = x^{\frac{2}{3}} + 5x^{-6} \Rightarrow F(x) = \frac{3}{5}x^{\frac{5}{3}} - x^{-5} = \frac{3}{5}\sqrt[3]{x^5} - \frac{1}{x^5}