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

color(blue)(I_1=int(csin(x)+dcos(x))/(csin(x)+dcos(x))dx=x+C_1

color(blue)(I_2=int(c cos(x)-dsin(x))/(csin(x)+dcos(x))dx=ln(abs(csin(x)+dcos(x)))+C_2

Can we determinate some constants A and B, such that

I=AI_1+BI_2

Then

asin(x)+bcos(x)
=A(csin(x)+dcos(x))+B(c cos(x)-dsin(x))

color(red)(a)sin(x)+color(green)(b)cos(x)=color(red)((Ac-Bd))sin(x)+color(green)((Ad+Bc))cos(x)

Solving for A and B

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