integrate 2cosx-3sinx/6cosx+4sinx using substitution method .

Asked by NAKSHATRA | 4th Nov, 2014, 08:58: AM

Expert Answer:

L e t space I equals integral fraction numerator 2 cos x minus 3 sin x over denominator 6 cos x plus 4 sin x end fraction d x T o space e v a l u a t e space t h i s space t y p e space o f space i n t e g r a l s comma space l e t space u s space u s e space t h e space f o l l o w i n g space a l g o r i t h m : A s s u m e comma N u m e r a t o r equals p open parentheses D i f f e r e n t i a t i o n space o f space d e n o m i n a t o r close parentheses plus q open parentheses D e n o m i n a t o r close parentheses rightwards double arrow minus 3 sin x plus 2 cos x equals p open parentheses 4 cos x minus 6 sin x close parentheses plus q open parentheses 4 sin x plus 6 cos x close parentheses rightwards double arrow minus 3 sin x plus 2 cos x equals open parentheses minus 6 p plus 4 q close parentheses sin x plus open parentheses 4 p plus 6 q close parentheses cos x C o m p a r i n g space t h e space c o e f f i c i e n t s comma space w e space h a v e comma minus 6 p plus 4 q equals minus 3 4 p plus 6 q equals 2 S o l v i n g space t h e space a b o v e space s i m u l tan e o u s space e q u a t i o n s comma space w e space h a v e comma space p equals 1 half space a n d space q equals 0 T h u s comma I equals integral fraction numerator p open parentheses 4 cos x minus 6 sin x close parentheses plus q open parentheses 4 sin x plus 6 cos x close parentheses over denominator 6 cos x plus 4 sin x end fraction d x rightwards double arrow I equals 1 half integral fraction numerator open parentheses 4 cos x minus 6 sin x close parentheses over denominator 6 cos x plus 4 sin x end fraction d x L e t space t equals 6 cos x plus 4 sin x ; space d t equals open parentheses minus 6 sin x plus 4 cos x close parentheses d x rightwards double arrow I equals 1 half integral fraction numerator d t over denominator t end fraction rightwards double arrow I equals 1 half log t plus C rightwards double arrow I equals 1 half log open parentheses 6 cos x plus 4 sin x close parentheses plus C

Answered by Vimala Ramamurthy | 4th Nov, 2014, 10:15: AM