Question

$\int \sqrt{1+sinxdx}$

Answer 1

Let I = $\int \sqrt{1+sinxdx}$

= $\int \sqrt{si{n}^2\frac{x}{2}+co{s}^2\frac{x}{2}+2sin\frac{x}{2}.cos\frac{x}{2}}dx[\because si{n}^2\frac{x}{2}+co{s}^2\frac{x}{2}=1]=\int \sqrt{{(sin\frac{x}{2}+cos\frac{x}{2})}^2}dx=\int |sin\frac{x}{2}+cos\frac{x}{2}|dx=-cos\frac{x}{2}.2+sin\frac{x}{2}.2+C=-2cos\frac{x}{2}+2sin\frac{x}{2}+C,whensin\frac{x}{2}+cos\frac{x}{2}\ge 0=2cos\frac{x}{2}-2sin\frac{x}{2}+C,whensin\frac{x}{2}+cos\frac{x}{2}<0$