𝑭ó𝒓𝒎𝒖𝒍𝒂𝒔 𝒅𝒆 𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒄𝒊ó𝒏 0. ∫ 𝑥 𝑛 𝑑𝑥 = 𝑥 𝑛+1 +𝑘 𝑛 +1 1. ∫ 𝑎 𝑑𝑥 = 𝑎𝑥 + 𝑘 2. ∫ 𝑎 𝑢 𝑑𝑥 = 𝑎 ∫ 𝑢 𝑑𝑥 3. ∫ 𝑢´ 𝑢 𝑛 𝑑𝑥 = 4. ∫ 𝑢 𝑛+1 + 𝑘 𝑝𝑎𝑟𝑎 𝑛 ≠ −1 𝑛+1 𝑢´ 𝑑𝑥 = ln(𝑢) + 𝑘 𝑢 5. ∫(𝑢 + 𝑣 − 𝑤)𝑑𝑥 = ∫ 𝑢 𝑑𝑥 + ∫ 𝑣 𝑑𝑥 − ∫ 𝑤 𝑑𝑥 6. ∫ 𝑢´ 𝑒 𝑢 𝑑𝑥 = 𝑒 𝑢 + 𝑘 7. ∫ 𝑢´ 𝑎𝑢 𝑑𝑥 = 𝑎𝑢 +𝑘 ln 𝑎 8. ∫ 𝑢´ sin(𝑢) 𝑑𝑥 = − cos(𝑢) + 𝑘 9. ∫ 𝑢´ cos(𝑢) 𝑑𝑥 = sin(𝑢) + 𝑘 𝑭ó𝒓𝒎𝒖𝒍𝒂𝒔 𝒅𝒆 𝑫𝒆𝒓𝒊𝒗𝒂𝒄𝒊ó𝒏 𝑰𝒅𝒆𝒏𝒕𝒊𝒅𝒂𝒅𝒆𝒔 𝑭𝒖𝒏𝒅𝒂𝒎𝒆𝒏𝒕𝒂𝒍𝒆𝒔 𝑑 𝑛 𝑑 𝑢 = 𝑛(𝑢 𝑛−1) (𝑢) 𝑑𝑥 𝑑𝑥 𝑑 𝑑 𝑑 (𝑢𝑣) = 𝑢 (𝑣) + 𝑣 (𝑢) 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑 𝑑 (𝑢) − 𝑢 𝑣 (𝑣) 𝑑 𝑢 𝑑𝑥 ( ) = 𝑑𝑥 2 𝑑𝑥 𝑣 𝑣 𝑑 𝑑 (𝑠𝑒𝑛 𝑢) = cos 𝑢 (𝑢) 𝑑𝑥 𝑑𝑥 𝑑 𝑑 (cos 𝑢) = −𝑠𝑒𝑛 𝑢 (𝑢) 𝑑𝑥 𝑑𝑥 𝑑 𝑑 (tan 𝑢) = sec 2 𝑢 (𝑢) 𝑑𝑥 𝑑𝑥 𝑑 𝑑 (𝑐𝑡𝑔 𝑢) = csc 2 𝑢 (𝑢) 𝑑𝑥 𝑑𝑥 10. ∫ 𝑢´ tan(𝑢) 𝑑𝑥 = − ln(cos(𝑢)) + 𝑘 𝑑 𝑑 (sec 𝑢) = sec 𝑢 tan 𝑢 (𝑢) 𝑑𝑥 𝑑𝑥 11. ∫ 𝑢´ cot(𝑢) 𝑑𝑥 = ln(sin (𝑢)) + 𝑘 𝑑 𝑑 (csc 𝑢) = − csc 𝑢 𝑐𝑡𝑔 𝑢 (𝑢) 𝑑𝑥 𝑑𝑥 12. ∫ 𝑢´ sec(𝑢) 𝑑𝑥 = ln(sec(𝑢) + tan (𝑢)) + 𝑘 13. ∫ 𝑢´ csc(𝑢) 𝑑𝑥 = ln(csc(𝑢) − cot(𝑢)) + 𝑘 14. ∫ 𝑢´ sec 2 (𝑢)𝑑𝑥 = tan(𝑢) + 𝑘 15. ∫ 𝑢´ csc 2 (𝑢)𝑑𝑥 = − cot(𝑢) + 𝑘 16. ∫ 𝑢´ sec(u) tan(u) dx = sec(u) + k 17. ∫ 𝑢´ csc(𝑢) cot(𝑢) 𝑑𝑥 = − csc(𝑢) + 𝑘 𝑢´ 1 𝑢 18. ∫ 2 𝑑𝑥 = arctan ( ) + 𝑘 𝑢 + 𝑎2 𝑎 𝑎 𝑢´ 1 𝑢−𝑎 19. ∫ 2 𝑑𝑥 = ln ( )+𝑘 𝑢 − 𝑎2 2𝑎 𝑢+𝑎 𝑢´ 1 𝑎+𝑢 20. ∫ 2 𝑑𝑥 = ln ( )+𝑘 2 𝑎 −𝑢 𝑎 𝑎−𝑢 𝑢´ 𝑢 21. ∫ 𝑑𝑥 = arcsin ( ) + 𝑘 2 2 𝑎 √𝑎 − 𝑢 𝑢´ 22. ∫ 𝑑𝑥 = ln (𝑢 + √𝑢 2 + 𝑎2 ) + 𝑘 √𝑢 2 + 𝑎2 𝑢´ 23. ∫ 𝑑𝑥 = ln (𝑢 + √𝑢 2 − 𝑎2 ) + 𝑘 √𝑢 2 − 𝑎2 𝑢´ 1 𝑢 24. ∫ 𝑑𝑥 = arcsec ( ) + 𝑘 𝑎 𝑎 𝑢√𝑢 2 − 𝑎2 1 𝑎 + √𝑢 2 + 𝑎 2 25. ∫ 𝑑𝑥 = − ln ( )+𝑘 2 2 𝑎 𝑢 𝑢√𝑢 + 𝑎 𝑢´ 1 𝑎 + √𝑎 2 − 𝑢 2 26. ∫ 𝑑𝑥 = − ln ( )+𝑘 𝑎 𝑢 𝑢√𝑎 2 − 𝑢 2 𝑑 1 𝑑 (𝑎𝑟𝑐 𝑆𝑒𝑛 𝑢) = (𝑢) 𝑑𝑥 √1 − 𝑢 2 𝑑𝑥 𝑢 𝑎2 𝑢 √𝑎 2 − 𝑢2 + arcsin ( ) + 𝑘 2 2 𝑎 𝑢 𝑎2 28. ∫ 𝑢´√𝑢 2 ± 𝑎 2 𝑑𝑥 = √𝑢 2 ± 𝑎 2 ± ln (𝑢 + √𝑢 2 ± 𝑎 2 ) + 𝑘 2 2 𝑐𝑠𝑐(𝑥) = 1 𝑠𝑒𝑛(𝑥) sec(𝑥) = 1 cos(𝑥) tan(𝑥) = 𝑠𝑒𝑛(𝑥) cos(𝑥) cot(𝑥) = 1 tan (𝑥) 𝑫𝒆𝒍 𝒕𝒆𝒐𝒓𝒆𝒎𝒂 𝒅𝒆 𝑷𝒊𝒕á𝒈𝒐𝒓𝒂𝒔 𝑠𝑒𝑛2 (𝑥) + cos 2 (𝑥) = 1 1 + tan2(𝑥) = sec 2 (𝑥) 1 + cot 2 (𝑥) = csc2 (𝑥) 𝑺𝒖𝒎𝒂𝒔 𝒚 𝒓𝒆𝒔𝒕𝒂𝒔 𝒅𝒆 á𝒏𝒈𝒖𝒍𝒐𝒔 𝑠𝑒𝑛(𝑥 + 𝑦) = 𝑠𝑒𝑛(𝑥)𝑐𝑜𝑠(𝑦) + 𝑐𝑜𝑠(𝑥)𝑠𝑒𝑛(𝑦) 𝑠𝑒𝑛(𝑥 − 𝑦) = 𝑠𝑒𝑛(𝑥)𝑐𝑜𝑠(𝑦) − 𝑐𝑜𝑠(𝑥)𝑠𝑒𝑛(𝑦) cos(𝑥 + 𝑦) = cos(𝑥) cos(𝑦) − 𝑠𝑒𝑛(𝑥)𝑠𝑒𝑛(𝑦) cos(𝑥 − 𝑦) = cos(𝑥) cos(𝑦) + 𝑠𝑒𝑛(𝑥)𝑠𝑒𝑛(𝑦) tan(𝑥 + 𝑦) = tan(𝑥) + tan(𝑦) (1 − tan(𝑥) tan(𝑦)) tan(𝑥 − 𝑦) = tan(𝑥) − tan(𝑦) (1 + tan(𝑥) tan(𝑦)) 𝑳𝒆𝒚 𝒅𝒆 𝒔𝒆𝒏𝒐𝒔 𝑠𝑒𝑛 𝐴 𝑠𝑒𝑛 𝐵 𝑠𝑒𝑛 𝐶 = = 𝑎 𝑏 𝑐 𝑳𝒆𝒚 𝒅𝒆 𝒄𝒐𝒔𝒆𝒏𝒐𝒔 𝑑 −1 𝑑 (𝑎𝑟𝑐 𝐶𝑜𝑠 𝑢) = (𝑢) 𝑑𝑥 √1−𝑢 2 𝑑𝑥 𝑐 2 = 𝑎 2 + 𝑏2 − 2𝑎𝑏 𝑐𝑜𝑠(𝐶) 𝑑 1 𝑑 (𝑎𝑟𝑐 𝑇𝑎𝑛 𝑢) = (𝑢) 2 𝑑𝑥 1 + 𝑢 𝑑𝑥 𝑠𝑒𝑛(−𝑥) = −𝑠𝑒𝑛(𝑥) 𝑑 −1 𝑑 (𝑎𝑟𝑐 𝑐𝑡𝑔 𝑢) = (𝑢) 𝑑𝑥 1 + 𝑢 2 𝑑𝑥 𝑻𝒓𝒂𝒔𝒍𝒂𝒄𝒊𝒐𝒏𝒆𝒔 cos(−𝑥) = cos(𝑥) 𝜋 𝑠𝑒𝑛 ( − 𝑥) = cos(𝑥) 2 𝜋 𝑡𝑎𝑛 ( − 𝑥) = 𝑐𝑜𝑡(𝑥) 2 tan(−𝑥) = − tan(𝑥) 𝜋 𝑐𝑜𝑠 ( − 𝑥) = 𝑠𝑒𝑛(𝑥) 2 𝑑 1 𝑑 (𝑎𝑟𝑐 𝑆𝑒𝑐 𝑢) = (𝑢) 2 𝑑𝑥 𝑑𝑥 𝑢√𝑢 − 1 𝑴ú𝒍𝒕𝒊𝒑𝒍𝒐𝒔 𝒅𝒆 á𝒏𝒈𝒖𝒍𝒐𝒔 𝑑 −1 𝑑 (𝑎𝑟𝑐 𝐶𝑠𝑐 𝑢) = (𝑢) 2 𝑑𝑥 𝑑𝑥 𝑢√𝑢 − 1 𝑐𝑜𝑠(2𝑥) = 𝑐𝑜𝑠 2 (𝑥) − 𝑠𝑒𝑛2 (𝑥) 𝑑 1 𝑑 (𝐿𝑛 𝑢) = (𝑢) 𝑑𝑥 𝑢 𝑑𝑥 𝑐𝑜𝑠(2𝑥) = 1 − 2𝑠𝑒𝑛2 (𝑥) 𝑑 𝑢 𝑑 (𝑎 ) = 𝑎𝑢 𝐿𝑛 𝑎 (𝑢) 𝑑𝑥 𝑑𝑥 𝑑 log 𝑒 𝑑 (log 𝑢) = (𝑢) 𝑑𝑥 𝑢 𝑑𝑥 𝑑 𝑢 𝑑 (𝑒 ) = 𝑒 𝑢 (𝑢) 𝑑𝑥 𝑑𝑥 𝑑 𝑑 𝑑 (𝑢 𝑣 ) = 𝑣 𝑢 𝑣−1 (𝑢) + (𝐿𝑛 𝑢)(𝑢𝑣 ) (𝑣) 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑢´ 27. ∫ 𝑢´ √𝑎 2 − 𝑢2 𝑑𝑥 = 𝑰𝒅𝒆𝒏𝒕𝒊𝒅𝒂𝒅𝒆𝒔 𝑻𝒓𝒊𝒈𝒐𝒏𝒐𝒎é𝒕𝒓𝒊𝒄𝒂𝒔 𝑪𝒐𝒎𝒑𝒆𝒕𝒂𝒓 𝑻𝒓𝒊𝒏𝒐𝒎𝒊𝒐𝟐 𝑷𝒆𝒓𝒇𝒆𝒄𝒕𝒐 𝑏 2 𝑏2 𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑥 + ) + (𝑐 − ) 2 4 𝑠𝑒𝑛(2𝑥) = 2 𝑠𝑒𝑛(𝑥) 𝑐𝑜𝑠(𝑥) 𝑐𝑜𝑠(2𝑥) = 2 𝑐𝑜𝑠 2 (𝑥) − 1 𝑥 1 − cos(𝑥) = 2𝑠𝑒𝑛2 (2 ) 𝑡𝑎𝑛(2𝑥) = 𝑡𝑎𝑛(𝑥) 1 − 𝑡𝑎𝑛2 (𝑥) 𝑠𝑒𝑛2 (𝑥) = 1 1 − 𝑐𝑜𝑠(2𝑥) 2 2 𝑐𝑜𝑠 2 (𝑥) = 1 1 + 𝑐𝑜𝑠(2𝑥) 2 2 𝑠𝑒𝑛(2𝑥) = 𝑠𝑖𝑛(𝑥)𝑐𝑜𝑠(𝑥) 2 𝑶𝒕𝒓𝒂𝒔 𝑥 𝑡𝑎𝑛 ( ) = 𝑐𝑠𝑐(𝑥) − 𝑐𝑜𝑡(𝑥) 2 𝑰𝒏𝒕𝒆𝒈𝒓𝒂𝒄𝒊ó𝒏 𝒑𝒐𝒓 𝒑𝒂𝒓𝒕𝒆𝒔 ∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢 JUAN CARLOS PRECIADO GÁMEZ 𝐶á𝑙𝑐𝑢𝑙𝑜 𝑑𝑒 á𝑟𝑒𝑎𝑠 𝑏 ∫ 𝑓(𝑥) 𝑑𝑥 = # 𝑎 𝑆𝑖 𝑙𝑎 𝑔𝑟𝑎𝑓𝑖𝑐𝑎 𝑠𝑒 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑎 𝑐𝑜𝑛 𝑒𝑙 𝑒𝑗𝑒 𝑥: 𝑠𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑛 𝑑𝑜𝑠 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙𝑒𝑠 𝑝 𝑏 ∫ 𝑓(𝑥) 𝑑𝑥 + ∫ 𝑓(𝑥) 𝑑𝑥 + 𝑎 𝑝 JUAN CARLOS PRECIADO GÁMEZ