Question

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

Answer 1

`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`