Forule For Stndrd Forule Of Integrls: u Integrl Clculus By OP Gupt [Indir Awrd Winner, +9-965 35 48] A B C D n n k, n n log k k log e e k k E sin cos k F cos sin G tn log sec k OR log cos k H cot log sin k OR log cosec k I sec log sec tn k OR log tn k 4 J cosec log cosec cot k OR log tn k K L sec cosec tn k cot k M sec tn sec c N cosec cot cosec k O P Q R S T U sec k tn k log log k k log k log k sin k V W X log k log k sin k Y log k, where is ny constnt (nd oviously, k is the integrl constnt) Z k, where is constnt (nd, k is the integrl constnt) List Of Forule By OP Gupt Pge - [] wwwtheopguptwordpressco
List Of Forule for Clss XII By OP Gupt (Electronics & Counictions Engineering) Methods of Integrtion: Though there is no generl ethod for finding the integrl of function, yet here we hve considered the following ethods sed on oservtions for evluting the integrl of function: ) Integrtion y Sustitution Method In this ethod we chnge the integrl f, where independent vrile is, to nother integrl in which independent vrile is t (sy) different fro such tht nd t re relted y g t Let u f then, f Agin s du dt g t so, we hve gt du du Now f gt dt dt On integrting oth sides wrt t, we get du dt f g t dt dt u f g t g t dt ie, f f g t g t dt where So, it is cler tht sustituting g t g t in f will give us the se result s otined y putting g t in plce of nd gtdt in plce of ) Integrtion y Prtil Frctions Consider f g defines rtionl polynoil function If the degree of nuertor ie f is greter thn or equl to the degree of denointor ie g then, this type of rtionl function is clled n iproper rtionl function And if degree of f is sller thn the degree of denointor ie g then, this type of rtionl function is clled proper rtionl function In rtionl polynoil functions if the degree (ie highest power of the vrile) of nuertor (Nr) is greter thn or equl to the degree of denointor (Dr), then (without ny dout) lwys perfor the division ie, divide the Nr y Dr efore doing nything nd therefter use the following: Nuertor Reinder Quotient Denointor Denointor On doing this, the rtionl function is resolved into prtil frctions The tle shown elow lists the types of sipler prtil frctions tht re to e ssocited with vrious kinds of rtionl functions which will e delt in our current study: TABLE DEMONSTRATING PARTIAL FRACTIONS OF VARIOUS FORMS For of the Rtionl Function For of the Prtil Frction p q, A B p q A B p q r c A B C c p q r A B C p q r c A B C c where c cn t e fctorized further List Of Forule By OP Gupt Pge - [] wwwtheopguptwordpressco
MATHEMATICS List Of Forule for Clss XII By OP Gupt (+9-9653548) c) Integrl By Prts If nd V e two functions of then, d V I II V V In finding integrls y this ethod, proper choice of functions nd V is crucil Though there is no fied rule for tking nd V (their choice is possile y prctice) yet, following rule is found to e quite helpful in deciding the functions nd V : If nd V re of different types, tke tht function s which coes first in the word ILATE Here I stnds for Inverse trigonoetricl function, L stnds for Logrithic function, A stnds for Algeric function, T stnds for Trigonoetricl function nd E stnds for the Eponentil function If oth the functions re trigonoetricl, tke tht function s V whose integrl is esier If oth the functions re lgeric, tke tht function s whose differentition is esier Soe integrnds re such tht they re not product of two functions Their integrls y e found y integrls y prts tking s the second function Logrithic nd inverse trigonoetric functions re eples of such functions The result of integrl ojective type questions Mking the Perfect Squre: STEP Consider the epression STEP Mke the coefficient of epression will look like, STEP3 Add nd sutrct e f ( ) f ( ) e f ( ) k cn e directly pplied in cse of the c s unity ie, c STEP4 The perfect squre of c y tking coon, fter doing so the originl to the epression otined in STEP s depicted here ie, c will e Vrious Integrl fors: p q Integrls of the for, c c p q c, p q c : Epress the d nuertor p q s shown here, ie, p q A c B Then on, otin the vlues of A nd B y equting the coefficients of like powers of nd constnts ters on oth the sides Then, integrte it d fter replcing p q y A c B using the vlues of A nd B sin cos Integrls of the for d : Epress Nuertor A Denointor BDenointor csin d cos Then otin the vlues of A nd B y equting the coefficients of sin nd cos on oth the sides nd proceed sin cos c Integrls of the for : Note tht the previous integrl for cn e considered s psin qcos r d specil cse of this for Epress Nuertor A Denointor BDenointor C Then otin the List Of Forule By OP Gupt Pge - [3] wwwtheopguptwordpressco
List Of Forule for Clss XII By OP Gupt (Electronics & Counictions Engineering) vlues of unknowns ie, A, B nd C y equting the coefficients of sin, cos nd the constnt ters on oth the sides nd hence proceed Integrls of the for sin cos, sin, cos, nd sin cos sin ccos : Divide the Nr nd Dr oth y nd then put tn t nd proceed cos Replce sec, if ny, in Dr y tn Integrls of the for sin cos, sin, cos nd sin cos : tn tn Use sin nd/ or cos Replce tn in the Nr y sec nd then put tn tn tn t nd then fter proceed Integrls of the for where M nd N re liner or qudrtic epressions in : M N M N Sustitutions Liner Liner t Qudrtic Liner t N N Liner Qudrtic t M Qudrtic Qudrtic N t or t M A Few Useful Quickies: ) c) n n f f f k, n n f f n f n n k ) f log f k f d) n n k n Forule & Properties Of Definite Integrls: P f F F F P f f P3 f f t dt P4, f f f List Of Forule By OP Gupt Pge - [4] wwwtheopguptwordpressco
MATHEMATICS List Of Forule for Clss XII By OP Gupt (+9-9653548) P5 P6 f f f f, if f is even function i e, f f P7 P8 f, if f is odd function i e, f f f f f f f / f, if f f, if f f Proof Of A Few Iportnt Properties: P4, f f f PROOF We know, F F F Consider F F F f (i) f (ii) And F F F f (iii) Adding the equtions (ii) nd (iii), we get F F f f Hence,, f [ By (i) f f f [HP] P5 PROOF Consider f f f Let t dt Also when t nd, when t So, f f tdt f tdt [By using P f t dt List Of Forule By OP Gupt Pge - [5] wwwtheopguptwordpressco
List Of Forule for Clss XII By OP Gupt (Electronics & Counictions Engineering) f f [Replcing t y, P3 Hence, f f [HP] SPECIAL CASE OF P5 Tke nd Then, f f The proof for the specil cse is se s is for the P5, so it hs een left s n eercise for you! P6 f f, if f is even function i e, f f, if f is odd function i e, f f PROOF We know tht Consider f Let t dt Also when t nd when t So, f f tdt f f f f (i) [By using P4 f tdt [By using P t dt f f dt [Replcing t y, P3 Therefore eqution (i) ecoes, f f f f f f, f, ie, f f if f f if f f, is even function f if f, if f is odd function P7 f f f f f [HP] List Of Forule By OP Gupt Pge - [6] wwwtheopguptwordpressco
MATHEMATICS List Of Forule for Clss XII By OP Gupt (+9-9653548) PROOF We know Consider f f f f (i) [By using P4 Let t dt Also when t nd when t So, f f tdt f tdt [By using P f t dt f f [Replcing t y, P3 So eqution (i) ecoes, f f f [HP], f if f f, if f f P8 f PROOF We know Consider f f f f (i) Let t dt Also when t nd when t So, f f tdt f tdt [By using P f t dt f f [Replcing t y, P3 So eqution (i) ecoes, f f f f f List Of Forule By OP Gupt Pge - [7] wwwtheopguptwordpressco
Hence, List Of Forule for Clss XII By OP Gupt (Electronics & Counictions Engineering) f, if f f f, if f f [HP] Definite integrl s the Liit Of A Su (First Principle Of Integrls): Tke tht function whose integrl vlue is to e clculted s f li ( ) f h f f h f h f n h h or, f li h f h f h f 3 h f nh h ie, f lih f rh n n r nd then use the given reltion,, such tht s n, h nd nh or, f lih f rh n n, such tht s n, h nd nh r Click on the following link to go for plesnt surprise: http://theopguptwordpressco/ths-rockers/ Hii, All! I hope this teture y hve proved eneficil for you While going through this teril, if you noticed ny error(s) or, soething which doesn t ke sense to you, plese ring it in y notice through SMS or Cll t +9-965 35 48 or Eil t theopgupt@gilco With lots of Love & Blessings! - OP Gupt Electronics & Counictions Engineering, Indir Awrd Winner wwwtheopguptwordpressco List Of Forule By OP Gupt Pge - [8] wwwtheopguptwordpressco