How to integrate question of types: int (asinx+bcosx)/(csinx+dcosx) dx where a,b,c, d are coefficients by several methods?
1 Answer
Apr 14, 2018
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