Kumpulan Rumus Lengkap Integral

Untuk lebih efektif dalam menyelesaikan soal-soal integral tidak akan terlepas dari kemampuan kita dalam menghafalkan rumus-rumus yang sudah dikenal dalam integral yaitu sebagai berikut.

BENTUK DASAR

1. $\int u^{n}du=\frac{1}{n+1}u^{n+1}+C,\, \textrm{jika}\, n\neq -1$

2. $\int u\, dv=uv-\int v\, du$

3. $\int \frac{1}{u}du=\ln \left|u \right|+C$

4. $\int e^{u}du=e^{u}+C$

5. $\int a^{u}du=\frac{a^{u}}{\ln a}+C$

6. $\int \sin u\, du=-\cos u+C$

7. $\int \cos u\, du=\sin u+C$

8. $\int \sec^{2} u\, du=\tan u+C$

9. $\int \csc^{2} u\, du=-\cot u+C$

10. $\int \sec u\, \tan u\, du=\sec u+C$

11. $\int \csc u \cot u\, du=-\csc u+C$

12. $\int \tan u\, du=\ln\left|\sec u \right|+C$

13. $\int \cot u\, du=\ln\left|\sin u \right|+C$

14. $\int \sec u\, du=\ln\left|\sec u +\tan u\right|+C$

15. $\int \csc u\, du=\ln\left|\csc u +\cot u\right|+C$

16. $\int \frac{du}{\sqrt{a^{2}-u^{2}}}=\sin^{-1}\frac{u}{a}+C$

17. $\int \frac{du}{a^{2}+u^{2}}=\frac{1}{a}\tan^{-1}\frac{u}{a}+C$

18. $\int \frac{du}{a^{2}-u^{2}}=\frac{1}{2a}\ln \left|\frac{u+a}{u-a} \right|+C$

19. $\int \frac{du}{u\sqrt{u^{2}-a^{2}}}=\frac{1}{a}\sec^{-1} \left|\frac{u}{a} \right|+C$

BENTUK TRIGONOMETRI

20. $\int \sin^{2}u\, du=\frac{1}{2}u-\frac{1}{4}\sin 2u+C$

21. $\int \cos^{2}u\, du=\frac{1}{2}u+\frac{1}{4}\sin 2u+C$

22. $\int \tan^{2}u\, du=\tan u-u+C$

23. $\int \cot^{2}u\, du=\cot u-u+C$

24. $\int \sin^{3}u\, du=-\frac{1}{3}\left ( 2+\sin^{2}u \right )\cos u+C$

25. $\int \cos^{3}u\, du=\frac{1}{3}\left ( 2+\cos^{2}u \right )\sin u+C$

26. $\int \tan^{3}u\, du=\frac{1}{2}\tan^{2}u+\ln \left|\cos u \right|+C$

27. $\int \cot^{3}u\, du=-\frac{1}{2}\cot^{2}u-\ln \left|\sin u \right|+C$

28. $\int \sec^{3}u\, du=\frac{1}{2}\sec u \tan u+\frac{1}{2}\ln \left|\sec u + \tan u\right|+C$

29. $\int \csc^{3}u\, du=-\frac{1}{2}\csc u \cot u+\frac{1}{2}\ln \left|\csc u + \cot u\right|+C$

30. $\int \sin au\sin bu\, du=\frac{\sin \left ( a-b \right )u}{2\left ( a-b \right )}-\frac{\sin\left ( a+b \right )u}{2\left ( a+b \right )}+C,\, \textrm{jika}\, a^{2}\neq b^{2}$

31. $\int \cos au\cos bu\, du=\frac{\sin \left ( a-b \right )u}{2\left ( a-b \right )}+\frac{\sin\left ( a+b \right )u}{2\left ( a+b \right )}+C,\, \textrm{jika}\, a^{2}\neq b^{2}$

32. $\int \sin au\cos bu\, du=-\frac{\cos \left ( a-b \right )u}{2\left ( a-b \right )}-\frac{\cos\left ( a+b \right )u}{2\left ( a+b \right )}+C,\, \textrm{jika}\, a^{2}\neq b^{2}$