1 = -stanix + 5 tan's + C In the above example only the part with partial fraction is substituted. For the integral part. Now we discuss an example in which Jointly used both the substitution method and the partial fraction method dx r + 3 gx + 3 / - Poy Example 15 d Assume 5- cos-p-4 sinp Suppose y = sinp

A functional equation is an equation, which relates the values assumed by a function at two or more points, which are themselves related in a particular manner. For example, we define an odd function by the relation f(-x) = -f(x) for all x.This defination can be paraphrased to say that it is a function f(x), which satisfies the functional relation f(x) + f(y) = 0, whenever x+y = 0. Of course this does not identify the function uniquely, sometimes with some additionl information, a function satisfying a given functional equation can be identified uniquely. Suppose a functional equation has a relation between f(x) and f(1/x) , then due to the reason that reciprocal of a reciprocal gives back the original number, we can substitute 1/x for x. This will result into another equation and solving these two, we can find f(x) uniquely. Similarly, we can solve an equation which contains f(x) and f(-x). Such equations are of repetitive nature . Suppose that for every x !=0 af(x)+ bf(1/x)=1/x-5 , where aneb then the value of the integral int_(1)^(2)f(x)dx is

Answer the given question based on the alphabet given below. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z If the letters of the word PRODUCTIVE which are at the odd-numbered position in the English alphabet are picked up and arranged in alphabetical order from left and if they are now substituted by Z, Y, X and so on, beginning from left, which letter will get substituted by W? 1) No letter will get substituted by W 2) O 3) E 4) C 5) None of these

The formula in which a certain integral involving some parameters in connected with some integrals of lower order is called a reduction formula. In most of the cases the reduction formula is obtained by the process of integrating by parts. Of course, in some cases the methods of differentiation are adopted.Now answer the question:If I_(m,p)=intx^m(a+bx^n)^pdx and I_(m,p)=(x^(m+1)(a+bx^n)^p)/(m+1)-f(m,n,p,b)I_(m+n,p-1) , then the value of f(1,2,3,4) is equal to (A) 8 (B) 10 (C) 12 (D) none of these

The formula in which a certain integral involving some parameters in connected with some integrals of lower order is called a reduction formula. In most of the cases the reduction formula is obtained by the process of integrating by parts. Of course, in some cases the methods of differentiation are adopted.Now answer the question:If I_(m-2,n+2)=intsin^(m-2)xcos^(n+2)xdx and I_(m,n)=-(sin^(m-1)xcos^(n+1)x)/(n+1)+f(m,n)I_((m-2),(n+2)) , then f(2,3) is equal to (A) 1/2 (B) 1/3 (C) 1/4 (D) 1/5

The formula in which a certain integral involving some parameters in connected with some integrals of lower order is called a reduction formula. In most of the cases the reduction formula is obtained by the process of integrating by parts. Of course, in some cases the methods of differentiation are adopted.Now answer the question:If I_(m-2,n+2)=intsin^(m-2)xcos^(n+2)xdx and I_(m,n)=-(sin^(m-1)xcos^(n+1)x)/(n+1)+f(m,n)I_((m-2),(n+2)) , then f(2,3) is equal to (A) 1/2 (B) 1/3 (C) 1/4 (D) 1/5

Answer the given question based on the alphabet given below. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z If the letters of the word ECO SURVEY which are at even-numbered positions in the English alphabet are picked up and arranged in alphabetical order from left and if they are now substituted by L, M, N, O and so on, which letter will get substituted by N? 1) S 2) R 3) V 4) No such letter 5) None of these

2 P J @ 8 $ L B 1 V # Q 6 delta G W 9 K C D 3 © • ₤ 5 F R 7 A Y 4. Four of the following five are similar in relation to their positions in the above sequence and hence form a group Which one does not belong to that group ?

Eight friends P, Q, R,S, T, V, X and Z are sitting around, a circular table facing centre. P sits third to the right of X and second to the left of R. S and T do not sit next to each other, S sits second to the right of Q, who is not a neighbour of P. T sits second to the left of V. What is the position of V with respect to Z in the above arrangement? A. Third to the right B. Fourth to the left C. Second to the right D. Fourth to the right 1) Only A 2) Both A and B 3) Both B and D 4) Only D 5) None of these

Alexander Williamson prepared diethyl by a simple method, now called as Williamson's ether synthesis . In this method, an alkyl halide is treated with sodium alkoxide prepared from sodium and alcohol. This reaction is used in the synthesis of symmetrical and unsymmetrical ethers. It may be noted taht for preparing unsymmetrical ethers, the halide used should preferably be primary because secondary and tertiary alkyl halides may form alkenes as major product due to elimination process. CH_(3)-underset(CH_(3))underset(|)overset(CH_(3))overset(|)(C)-Br+underset("Sodium ethoxide")(Na^(+)O^(-)-C_(2)H_(5))rarrunderset("2-methylpropene")(CH_(3)-overset(CH_(3))overset(|)(C)=CH_(2))+C_(2)H_(5)OH+NaBr Aryl ethers or phenolic ethers can be prepared by using sodium phenoxide and alkyl halides. However, aryl halides and sodium alkoxide cannot be used for preparing phenolic ethers because aryl halides are less reactive towards nucleophilic substitution reaction Methyl tertiary butyl ether (MTBE) is added in gasoline to improve its octane number . CH_(3)-underset(CH_(3))underset(|)overset(CH_(3))overset(|)(C)-O-CH_(3) Which of the following is the best method for synthesis of the above ether ?