## Vineet Loomba is teaching live on Unacademy Plus

DEFINITE INTEGRATION JEE MAIN AND ADVANCED IIT-IEE MATHS MADE EASY PREPARED BY: ER. VINEET LOOMBA IITiAN | IIT-JEE MENTOR FoR SURE SHoT SUcCESS IN JEE MAIN AND ADVANCED (IIT-JEE)

100% SUCCESS IN JEE MAIN AND ADVANCED(IIT-JEE ABOUT ME B.Tech. From IIT Roorkee -$ IIT-JEE Mentor Since 2010. -^ Youtube Channel with 10k Active Followers Founder @ vineetloomba.com * Follow me @ https://unacademy.com/user/vineetloomba to get unacademy apysaler d updates or search me on Google * Share among your peers as SHARING is CARING!! FoR SuRE SHOT SuccESS IN JEE MAIN AND ADvanceD (IIT-JEE)

100% SUCCESS IN JEE MAIN AND ADVANCED(IIT-JEE Other Detailed Courses Made so far on Unacademv: V Strategy for JEE Main and Advanced V Sets, Relations and Functions V Trigonometry V Applications of Derivatives Limits, Continuity. Differentiability V Indefinite Integration V Definite Integration Complex Number V Logarithmic Functions V Sequences Series V Most Important Questions in IIT-JEE Permutations Combinations Binomial Theorem V Straight Lines V Applications of Integrals MathematicS V Parabola (Detailed Course) v Inverse Trigonometry V Mathematical Reasoning V Ellipse (Detailed Course) V Hyperbola Circles (Detailed Course) V Probability Upcoming Courses next month: Statistics As per your Demand Differential Equations V Quadratic Equations FoR SURE SHOT SucCESS IN JEE MAIN AND ADVANCED (IIT-JEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Definite Integral as a Limit of Sum. Let f(x) be a continuous real valued function defined on the closed interval [a, b] which is divided into n parts as shown in figure The point of division on x-axis are a, a + h, a + 2h a + (n-1)h, a + nh, where Let S denotes the area of these n rectangles b-a = Clearly, Sn is area very close to the area of the region boundedaa-h af2n......ato-X by curve y f(x), X-axis and the ordinates x a, xb. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Hence | f(x) dx= Lt (b- a)r f a+ Note: We can also write n in MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Steps to express the limit of sum as definte integral Step 1. Replace-by x, _ by dx and n L. by f Step 2. Evaluatenn as lower and upper limits respectively. Lt by putting least and greatest values ofr pn For example n f(x) dx 0 Lt Lt p) r =np MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Ex. Evaluate : lim 2n Sol. Limit = lim lim 1.1 1 log 2. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Ex. Find the value of 2 (n +2 Sol. Here tr =(n-r)2-n.[14(r / n Therefore given series = lim 2 Now lower limit lim | upper limit = lim Given series - (10,2 d 2 MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Lt Example : Eva222 n2 +32. 5n ntr Sol 2 - 0, when r1, lower limit 0 n-n and Lt r 2, when r= 2n, upper limit 2 2x 1+x2 dx i 2 = tan-1x], loge (1 + x 2 ) | = tan-1 2 +-log,5 + MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) lim n+1 n+12equals 2 12 2 n+2 Ex. (A) +log2 (B) og2 ! (C) -2log 4 (D) None of these T- n+r .. given limitdx 01+x2 MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)