JEE Main & Advanced Mathematics Indefinite Integrals Integrals of the form \[\int{\frac{a sinx+b cosx}{c sinx+d cosx} }\]and \[\int{\frac{a\,sinx+bcos\,x+q}{c\,sinx+d\,cosx+r}}\]

(1) Integrals of the form\[\int{\frac{a sinx+b cos x}{c sinx+d\,\mathbf{cos} x}}\,dx\]: Such rational functions of \[\sin x\] and \[\cos x\] may be integrated by expressing the numerator of the integrand as follows :

Numerator \[=M\] (Diff. of denominator) \[+N\] (Denominator)

i.e., \[a\sin x+b\cos x=M\frac{d}{dx}(c\sin x+d\cos x)+N(c\sin x+d\cos x)\]

The arbitary constants \[M\] and \[N\] are determined by comparing the coefficients of \[\sin x\] and \[\cos x\] from two sides of the above identity. Then, the given integral is

\[I=\int{\frac{a\sin x+b\cos x}{c\sin x+d\cos x}}\,dx\]

\[=\int{\frac{M(c\cos x-d\sin x)+N(c\sin x+d\cos x)}{c\sin x+d\cos x}}\,dx\]

\[=M\int{\frac{c\cos x-d\sin x}{c\sin x+d\cos x}}\,dx+N\int{1dx}\]

\[=M\log |c\sin x+d\cos x|+Nx+c.\]

(2) Integrals of the form \[\int{\frac{a\,sinx+bcosx+q}{csinx+dcosx+r}}\,dx\] : To evaluate this type of integrals, we express the numerator as follows: Numerator \[=M\text{(Denominator)}+N\text{(Differentiation}\,\text{of}\,\text{denominator)}+P\]

i.e.,\[(c\sin x+b\cos x+q)=M(c\sin x+d\cos x+r)\] \[+N(c\cos x-d\sin x)+P.\]

where M, N, P  are constants to be determined by comparing the coefficients of \[\sin x,\,\cos x\] and constant term on both sides.

\[\therefore \,\,\int{\frac{a\sin x+b\cos x+q}{c\sin x+d\cos x+r}}\,dx\]

\[=\int{M\,dx}+N\int{\frac{\text{Diff}\text{.}\,\text{of}\,\text{denominator}}{\text{Denominator}}\,dx}\]\[+\int{\frac{dx}{c\sin x-d\cos x+r}}\]

\[=Mx+N\log |\text{Denominator}|\,\] \[+P\int{\frac{dx}{c\sin x+d\cos x+r}}\].