Logarithms and Integrals with Inverse Trigonometric Functions, Lecture notes of Differential and Integral Calculus

Download Logarithms and Integrals with Inverse Trigonometric Functions and more Lecture notes Differential and Integral Calculus in PDF only on Docsity! Logarithms ׬ .5 𝑑𝑢 𝑢 = ln 𝑢 + 𝐶, 𝑢 >0 ׬ .’5 𝑑𝑢 𝑢 = ln(−𝑢) + 𝐶, 𝑢 <0 ׬ .’’5 𝑑𝑢 𝑢 = ln 𝑢 + 𝐶, 𝑢 ≠0 2. Evaluate ׬ 𝑥2−𝑥 𝑥+1 𝑑𝑥 1 2 𝑥2 − 2𝑥 + 2 ln 𝑥 + 1 + 𝐶 pes axe “ ' ax ~ a _ LA A M 7] tH x ~ RK Ed in [x4] *C + 2B In lu 3. ׬ e3xdx 5+2e3x 1 6 ln 5 + 2e3x + C Examples: 1. { 5°* dx O=S Uc O% dus ax OU oo dey 2  ׬ .2 𝑥𝑒𝑥 2 𝑑𝑥 1 2 𝑒𝑥 2 + 𝐶 ׬ .4 sin 2𝑥 𝑒cos 2𝑥 𝑑𝑥 − 1 2 𝑒cos 2𝑥 + 𝐶 Integrals Giving Inverse Trigonometric Functions න 𝑑𝑢 𝑎2 − 𝑢2 = 𝑠𝑖𝑛−1 𝑢 𝑎 + 𝐶 න 𝑑𝑢 𝑎2 + 𝑢2 = 1 𝑎 𝑡𝑎𝑛−1 𝑢 𝑎 + 𝐶 න 𝑑𝑢 𝑢 𝑢2 − 𝑎2 = 1 𝑎 𝑠𝑒𝑐−1 𝑢 𝑎 + 𝐶 Examples: (S 4x2 — — oy == SK \ iF —_ = | J car A _— OU SGP=a? we -— 40 a 8  3. න 𝑑𝑥 2𝑥2 + 2𝑥 + 3 1 5 𝑡𝑎𝑛−1 2𝑥 + 1 5 + 𝐶 \ eae 3 aK oe 43 a Ate x* +) +3- a £ mw J alae Ly + S 5 \ “(oral * =| et = an (EY + (xa) 5 \ zX rT v 72 Ons ax m A , IKTAKAD > eo a4 we atk ae