How to integrate question of types: #int (asinx+bcosx)/(csinx+dcosx) dx# where a,b,c, d are coefficients by several methods?
1 Answer
#I=((ac+bd)x+(bc-ad)ln(abs(csin(x)+dcos(x))))/(c^2+d^2)+C#
Explanation:
We want to solve
#I=int(asin(x)+bcos(x))/(csin(x)+dcos(x))dx#
Notice the easier integrals
Can we determinate some constants
#I=AI_1+BI_2#
Then
Solving for
#A=(ac+bd)/(c^2+d^2)#
#B=(bc-ad)/(c^2+d^2)#
Thus
#I=(ac+bd)/(c^2+d^2)I_1+(bc-ad)/(c^2+d^2)I_2#
#color(white)(I)=((ac+bd)I_1+(bc-ad)I_2)/(c^2+d^2)#
#color(white)(I)=((ac+bd)x+(bc-ad)ln(abs(csin(x)+dcos(x))))/(c^2+d^2)+C#