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integrate 2sinx-cosx+3/sinx-2cosx+1 dx
Asked by ayush9828450465 | 15 Oct, 2020, 10:17: AM
Expert Answer
Given integration :-
Let u = sinx - 2cosx +1 , then du = ( cosx+2sinx ) dx
numerator of integrand :- (2 sinx - cosx +3 ) dx = ( 2sinx + cosx -2cosx +3 ) dx = du + (3 - 2 cosx) dx
Hence given intgration becomes
.....................(1)
Where
......................(2)
To integrate above, we use the substitutions , t = tan(x/2) , sinx = 2t / ( 1+t2 ) , cosx = (1-t2 )/(1+t2 ) , dx = 2dt / ( 1+ t2 )
By using the above substitutions, we get
.........................(3)
Integrand in the above integration is written as partial fraction as given below
...................(4)
Hence we write
(1+5 t2 ) = A ( 3t -1 ) ( 1 + t2 ) + B ( 1+t) (1 +t2 ) + ( C t + D ) ( 1 + t ) ( 3 t -1 ) ........................(5)
if we put t = 1/3 in above eqn.(5) , we get B = 21/20
if we put t = -1 in above eqn.(5) , we get A = -3/4
if we put t = 0 in above eqn.(5) , we get 1 = -A +B -D = (3/4) + 21/20 -D , hence D = 4/5
if we put t = 1 in above eqn.(5) , we get 6 = 4A +4B +( C+D)4 , By substituting values for A , B and D , we get C = 2/5
using the values A, B, C and D , we rewrite integration as given in eqn.(3) as
Integration of each term in above equation is known .
Hence after integration of each term , by using substitution t = tan(x/2) and after simplification , we get
Hence using eqn.(1) , given integration becomes
Answered by Thiyagarajan K | 15 Oct, 2020, 04:00: PM
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