INTEGRAL QUESTIONS AND DISCUSSION

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Integral Basic Formula

This basic formula is obtained from the basic differential formula.
1. $$\int~ax^n~dx=\frac{a}{n+1}x^{n+1}+C$$ 2. $$\int~[f(x)]^n~d(f(x))=\frac{1}{n+1}.[f(x)]^{n+1}+C$$ 3. $$\int~\frac{d(f(x))}{f(x)}=\text{ln}[f(x)]+C$$ 4. $$\int~e^u~du=e^u+C$$ 5. $$\int~\text{sin}x~dx=-\text{cos}x+C$$ 6. $$\int~\text{cos}x~dx=\text{sin}x+C$$ 7. $$\int~\text{sec}^2x~dx=\text{tan}x+C$$ 8. $$\int~\text{csc}^2x~dx=-\text{cot}x+C$$ 9. $$\int~\text{tan}x.\text{sec}x~dx=\text{sec}x+C$$ 10. $$\int~\text{cot}x.\text{csc}x~dx=-\text{csc}x+C$$ 11. Partial Integral: $$\int~u~dv=u.v-\int~v~du$$ 12. Reimann Integral: $$\int_{b}^{a}~f'(x)~dx=f(a)-f(b)$$


Questions and Discussion

1. $$\int~(-2x^{-3}+\sqrt{x})~dx=…$$

2. $$\int~(\sqrt[3]{x^2}-\sqrt{5x^3}+9)~dx=…$$

3. $$\int~\sqrt[5]{3x-8}~dx=…$$

4. $$\int~5x\sqrt{x^2+4}~dx=…$$

5. $$\int~\text{sin}^3x.\text{cos}x~dx=…$$


6. $$\int~-10x.\text{cos}(x^2+5)~dx=…$$

7. $$\int~e^{2x+5}~dx=…$$

8. $$\int~-13^{-x+1}~dx=…$$

9. $$\int~\text{cot}x~dx=…$$

10. $$\int~\frac{dx}{\sqrt{x^2-4}}=…$$


11. $$\int~\text{ln}x~dx=…$$

12. $$\int~x.\text{sin}^2x~dx=…$$

13. $$\int~\frac{dx}{x^2-1}$$

14. $$\int~2x.\sqrt{x+5}~dx=…$$

15. $$\int~\text{log}(4x^2)~dx=…$$


16. $$\int_{1}^{3}~(4x^2-2x+1)~dx=…$$

17. The area delimited by the curve $ y = x ^ 2-6x + 5 $ and the $ x $ axis is …. unit area.


18. Given the path of the parabola $ y = -x ^ 2 + 3x-2 $. If a particle moves on a trajectory from point $ (0, ~ -2) $ to point $ (2, ~ 0) $, then the length of the path passed through that particle is …. a unit of length.

19. The volume of the rotating object rotated over the line $ x = 3 $ and is limited by $ y = x ^ 2-6x + 5 $ and the $ x $ axis is …. the unit of volume.

20. The area of ​​the ellipse $ 9x ^ 2 + 4y ^ 2 = 36 $ is …. units of area.


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