Calculus 2 Cheat Sheet Formulas, Cheat Sheet of Differential and Integral Calculus

Download Calculus 2 Cheat Sheet Formulas and more Cheat Sheet Differential and Integral Calculus in PDF only on Docsity! Calculus 2 Long Exam 2 Formulas Integrals yielding to trig functions: 1. ∫ 𝑑𝑢 √1−𝑢2 = sin−1 𝑢 + 𝐶 2. ∫ 𝑑𝑢 𝑢2+1 = tan−1 𝑢 + 𝐶 3. ∫ 𝑑𝑢 𝑢√𝑢2−1 = sec−1 𝑢 + 𝐶 4. ∫ 𝑑𝑢 √𝑎2−𝑢2 = sin−1 𝑢 𝑎 + 𝐶 5. ∫ 𝑑𝑢 𝑢2+𝑎2 = 1 𝑎 tan−1 𝑢 𝑎 + 𝐶 6. ∫ 𝑑𝑢 𝑢√𝑢2−𝑎2 = 1 𝑎 sec−1 𝑢 𝑎 + 𝐶 Integration of hyperbolic functions: 1. ∫ sinh 𝑢 𝑑𝑢 = cosh 𝑢 + 𝐶 2. ∫ cosh 𝑢 𝑑𝑢 = sinh 𝑢 + 𝐶 3. ∫ sech2 𝑢 𝑑𝑢 = tanh 𝑢 + 𝐶 4. ∫ csch2 𝑢 𝑑𝑢 = −coth 𝑢 + 𝐶 5. ∫ sech 𝑢 tanh 𝑢 𝑑𝑢 = −sech 𝑢 + 𝐶 6. ∫ csch 𝑢 coth 𝑢 𝑑𝑢 = −csch 𝑢 + 𝐶 Properties of hyperbolic functions:  sinh 𝑥 = 𝑒𝑥−𝑒−𝑥 2  cosh 𝑥 = 𝑒𝑥+𝑒−𝑥 2  tanh 𝑥 = 𝑒𝑥−𝑒−𝑥 𝑒𝑥+𝑒−𝑥  coth 𝑥 = 𝑒𝑥+𝑒−𝑥 𝑒𝑥−𝑒−𝑥  sech 𝑥 = 2 𝑒𝑥+𝑒−𝑥  csch 𝑥 = 2 𝑒𝑥−𝑒−𝑥  cosh2 𝑥 − sinh2 𝑥 = 1  1 − tanh2 𝑥 = sech2 𝑥  1 − coth2 𝑥 = −csch2 𝑥  cosh 𝑥 + sinh 𝑥 = 𝑒𝑥  cosh 𝑥 − sinh 𝑥 = 𝑒−𝑥  sinh(𝑥 + 𝑦) = sinh 𝑥 cosh 𝑦 + cosh 𝑥 sinh 𝑦  cosh(𝑥 + 𝑦) = cosh 𝑥 cosh 𝑦 + sinh 𝑥 sinh 𝑦  sinh 2𝑥 = 2 sinh 𝑥 cosh 𝑥  cosh 2𝑥 = cosh2 𝑥 + sinh2 𝑥 7. ∫ 𝑑𝑢 √𝑢2+1 = sinh−1 𝑢 + 𝐶 8. ∫ 𝑑𝑢 √𝑢2−1 = cosh−1 𝑢 + 𝐶 ; 𝑢 > 1 9. ∫ 𝑑𝑢 1−𝑢2 = tanh−1 𝑢 + 𝐶 ; |𝑢| < 1 10. ∫ 𝑑𝑢 1−𝑢2 = coth−1 𝑢 + 𝐶 ; |𝑢| > 1 11. ∫ 𝑑𝑢 √𝑢2+𝑎2 = sinh−1 𝑢 𝑎 + 𝐶 12. ∫ 𝑑𝑢 √𝑢2−𝑎2 = cosh−1 𝑢 𝑎 + 𝐶 13. ∫ 𝑑𝑢 𝑎2−𝑢2 = 1 a tanh−1 𝑢 𝑎 + 𝐶 ; |𝑢| < 𝑎 14. ∫ 𝑑𝑢 𝑎2−𝑢2 = 1 a coth−1 𝑢 𝑎 + 𝐶 ; |𝑢| > 𝑎 Integration by parts: 1. 𝑢𝑣 − ∫ 𝑣 𝑑𝑢 Cases of integration: 1. ∫ sin𝑛 𝑥 𝑑𝑥 𝑜𝑟 ∫ cos𝑛 𝑥 𝑑𝑥 ; 𝑛 = +𝑜𝑑𝑑 a. ∫ sin𝑛 𝑥 𝑑𝑥 = (sin𝑛−1 𝑥)(sin 𝑥 𝑑𝑥) = (sin2 𝑥) 𝑛−1 2 (sin 𝑥)𝑑𝑥 = (1 − cos2 𝑥) 𝑛−1 2 (sin 𝑥)𝑑𝑥 b. ∫ cos𝑛 𝑥 𝑑𝑥 = (cos𝑛−1 𝑥)(cos 𝑥 𝑑𝑥) = (cos2 𝑥) 𝑛−1 2 (cos 𝑥)𝑑𝑥 = (1 − sin2 𝑥) 𝑛−1 2 (cos 𝑥)𝑑𝑥 2. ∫ sin𝑛 𝑥 cos𝑚 𝑥 𝑑𝑥 where one exponent is +odd a. ∫ sin𝑛 𝑥 cosm 𝑥 𝑑𝑥 = sin𝑛−1 𝑥 cos𝑚 𝑥 (sin 𝑥)𝑑𝑥 = (sin2 𝑥) 𝑛−1 2 cos𝑚 𝑥 (sin 𝑥)𝑑𝑥 = (1 − cos2 𝑥) 𝑛−1 2 cos𝑚 𝑥 (sin 𝑥)𝑑𝑥 b. ∫ sin𝑛 𝑥 cosm 𝑥 𝑑𝑥 = sin𝑛 𝑥 cos𝑚−1 𝑥 (cos 𝑥)𝑑𝑥 = sin𝑛 𝑥 (cos2 𝑥) 𝑚−1 2 (cos 𝑥) 𝑑𝑥 = sin𝑛 𝑥 (1 − sin2 𝑥) 𝑚−1 2 (cos 𝑥) 𝑑𝑥 3. ∫ sin𝑛 𝑥 cos𝑚 𝑥 𝑑𝑥 exponent is +even a. ∫ sin𝑛 𝑥 𝑑𝑥 = (sin2 𝑥) 𝑛 2 𝑑𝑥 = ( 1−cos 2𝑥 2 ) 𝑛 2 𝑑𝑥 b. ∫ cos𝑚 𝑥 𝑑𝑥 = (cos2 𝑥) 𝑚 2 𝑑𝑥 = ( 1+cos 2𝑥 2 ) 𝑚 2 𝑑𝑥 c. ∫ sin𝑛 𝑥 cosm 𝑥 𝑑𝑥 = (sin2 𝑥) 𝑛 2(cos2 𝑥) 𝑚 2 𝑑𝑥 = ( 1 − cos 2𝑥 2 ) 𝑛 2 ( 1 + cos 2𝑥 2 ) 𝑚 2 𝑑𝑥 4. ∫ tan𝑛 𝑥 𝑑𝑥 𝑜𝑟 ∫ cot𝑛 𝑥 𝑑𝑥 ; 𝑛 = +𝑖𝑛𝑡𝑒𝑔𝑒𝑟 a. ∫ tan𝑛 𝑥 𝑑𝑥 = tan𝑛−2 𝑥 tan2 𝑥𝑑𝑥 = tan𝑛−2 𝑥 (sec2 𝑥 − 1)𝑑𝑥 b. ∫ cot𝑛 𝑥 𝑑𝑥 = cot𝑛−2 𝑥 cot2 𝑥𝑑𝑥 = cot𝑛−2 𝑥 (csc2 𝑥 − 1)𝑑𝑥 5. ∫ sec𝑛 𝑥 𝑑𝑥 𝑜𝑟 ∫ csc𝑛 𝑥 𝑑𝑥 ; 𝑛 = +𝑒𝑣𝑒𝑛 a. ∫ sec𝑛 𝑥 𝑑𝑥 = sec𝑛−2 𝑥 sec2 𝑥𝑑𝑥 = (sec2 𝑥) 𝑛−2 2 𝑥 (sec2 𝑥)𝑑𝑥 = (tan2 𝑥 + 1) 𝑛−2 2 𝑥 (sec2 𝑥)𝑑𝑥 b. ∫ csc𝑛 𝑥 𝑑𝑥 = csc𝑛−2 𝑥 csc2 𝑥𝑑𝑥 = (csc2 𝑥) 𝑛−2 2 𝑥 (csc2 𝑥)𝑑𝑥 = (cot2 𝑥 + 1) 𝑛−2 2 𝑥 (csc2 𝑥)𝑑𝑥 6. ∫ tan𝑛 𝑥 sec𝑚 𝑥 𝑑𝑥 | ∫ cot𝑛 𝑥 csc𝑚 𝑥𝑑𝑥 m=+even a. ∫ tan𝑛 𝑥 sec𝑚 𝑥 𝑑𝑥 = tan𝑛 𝑥 sec𝑚−2 𝑥(sec2 𝑥)𝑑𝑥 = tan𝑛 𝑥 (sec2 𝑥) 𝑚−2 2 (sec2 𝑥)𝑑𝑥 = tan𝑛 𝑥 (tan2 𝑥 + 1) 𝑚−2 2 (sec2 𝑥)𝑑𝑥 b. ∫ cot𝑛 𝑥 𝑐sc𝑚 𝑥 𝑑𝑥 = cot𝑛 𝑥 csc𝑚−2 𝑥(csc2 𝑥)𝑑𝑥 = cot𝑛 𝑥 (csc2 𝑥) 𝑚−2 2 (csc2 𝑥)𝑑𝑥 = cot𝑛 𝑥 (cot2 𝑥 + 1) 𝑚−2 2 (csc2 𝑥)𝑑𝑥 7. ∫ tan𝑛 𝑥 sec𝑚 𝑥 𝑑𝑥 | ∫ cot𝑛 𝑥 csc𝑚 𝑥𝑑𝑥 n=+odd a. ∫ tan𝑛 𝑥 sec𝑚 𝑥 𝑑𝑥 = tan𝑛−1 𝑥 sec𝑚−1 𝑥 (𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥)𝑑𝑥 = (tan2 𝑥) 𝑛−1 2 sec𝑚−1 𝑥(𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥)𝑑𝑥 = (sec2 𝑥 − 1) 𝑛−1 2 sec𝑚−1 𝑥(𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥)𝑑𝑥 b. ∫ cot𝑛 𝑥 csc𝑚 𝑥 𝑑𝑥 = cot𝑛−1 𝑥 csc𝑚−1 𝑥 (𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥)𝑑𝑥 = (cot2 𝑥) 𝑛−1 2 csc𝑚−1 𝑥 (𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥)𝑑𝑥 = (csc2 𝑥 − 1) 𝑛−1 2 csc𝑚−1 𝑥 (𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥)𝑑𝑥 8. ∫ sec𝑛 𝑥 𝑑𝑥 𝑜𝑟 ∫ csc𝑛 𝑥 𝑑𝑥 ; 𝑛 = +𝑜𝑑𝑑 a. 𝑢 = sec𝑛−2 𝑥 & 𝑑𝑣 = sec2 𝑥 𝑑𝑥 b. 𝑢 = csc𝑛−2 𝑥 & 𝑑𝑣 = csc2 𝑥 𝑑𝑥 9. ∫ tan𝑛 𝑥 sec𝑚 𝑥 𝑑𝑥 | ∫ cot𝑛 𝑥 csc𝑚 𝑥𝑑𝑥 n=+even|m=+odd a. ∫ tan𝑛 𝑥 sec𝑚 𝑥 𝑑𝑥 = (tan2 𝑥) 𝑛 2 sec𝑚 𝑥𝑑𝑥 = (sec2 𝑥 − 1) 𝑛 2 sec𝑚 𝑥𝑑𝑥 b. ∫ cot𝑛 𝑥 csc𝑚 𝑥 𝑑𝑥 = (cot2 𝑥) 𝑛 2 csc𝑚 𝑥𝑑𝑥 = (csc2 𝑥 − 1) 𝑛 2 csc𝑚 𝑥𝑑𝑥