Let I=∫sin2x+sinx1+sinx+cosxdx, J=∫cos2x+cosx1+sinx+cosxdx and c is the constant of integration.FunctionIntegral(a) I (p) 12(x−sinx−cosx)+c (b) J (q) 12(x+sinx+cosx)+c (c) I + J (r) x+c (d) I - J (s) c−cosx−sinx (t) c+cosx+sinx (u) −12(x+sinx+cosx+c)Which of the following is the only CORRECT combination?

Let I=∫sin2x+sinx1+sinx+cosxdx, J=∫cos2x+cosx1+sinx+cosxdx and c i

1.	Let I=∫sin2x+sinx1+sinx+cosxdx, J=∫cos2x+cosx1+sinx+cosxdx and c is the constant of integration.FunctionIntegral(a) I (p) 12(x−sinx−cosx)+c (b) J (q) 12(x+sinx+cosx)+c (c) I + J (r) x+c (d) I - J (s) c−cosx−sinx (t) c+cosx+sinx (u) −12(x+sinx+cosx+c)Which of the following is the only CORRECT combination?
Answer» Let $I = \int \frac{{sin}^{2} x + sin x}{1 + sin x + cos x} d x$ , $J = \int \frac{{cos}^{2} x + cos x}{1 + sin x + cos x} d x$ and $c$ is the constant of integration. $\begin{matrix} Function & Integral \begin{matrix} (a) I \end{matrix} & (p) \frac{1}{2} (x - sin x - cos x) + c \begin{matrix} (b) J \end{matrix} & (q) \frac{1}{2} (x + sin x + cos x) + c \begin{matrix} (c) I + J \end{matrix} & (r) x + c \begin{matrix} (d) I - J \end{matrix} & (s) c - cos x - sin x \begin{matrix} \end{matrix} & (t) c + cos x + sin x \begin{matrix} \end{matrix} & (u) - \frac{1}{2} (x + sin x + cos x + c) \end{matrix}$ Which of the following is the only CORRECT combination?