Evaluate $\int_0^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} dx$
Answer
Let $u=a+b-x$. Then $du=-dx$ and
\[I=\int_a^{b}\frac{f(x)}{f(a+b-x)+f(x)}dx\]
\[=\int_{b}^{a}\frac{f(a+b-u)}{f(u)+f(a+b-u)}(-du)\]
\[=\int_{a}^{b}\frac{f(a+b-u)}{f(u)+f(a+b-u)}du\]
\[=\int_{a}^{b}\frac{f(a+b-x)}{f(x)+f(a+b-x)}dx=I.\]
Hence
\[2I=\int_a^{b}\frac{f(x)}{f(a+b-x)+f(x)}dx+\int_{a}^{b}\frac{f(a+b-x)}{f(x)+f(a+b-x)}dx\]
\[=\int_a^{b}\frac{f(x)+f(a+b-x)}{f(a+b-x)+f(x)}dx=\int_a^{b}1dx=b-a.\]
So
\[I=\int_a^{b}\frac{f(x)}{f(a+b-x)+f(x)}dx=\frac{b-a}{2}.\]
Now letting $f(x)=\sqrt{\sin x}$, $a=0, b=\frac{\pi}{2}$. We have
\[f(\frac{\pi}{2}+0-x)=\sqrt{\sin (\frac{\pi}{2}-x)}=\sqrt{\cos x}.\]
Hence
\[\int_0^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx=\int_0^{\frac{\pi}{2}}\frac{f(x)}{f(0+\frac{\pi}{2}-x)+f(x)}dx\]
\[\frac{\frac{\pi}{2}-0}{2}=\frac{\pi}{4}.\]
- answered
- 1171 views
- $20.00
Related Questions
- Given $|f(x) - f(y)| \leq M|x-y|^2$ , prove that f is constant.
- Show that the line integral $ \oint_C y z d x + x z d y + x y d z$ is zero along any closed contour C .
- Answer is done but need help
- Find the volume of the solid obtained by rotating $y=x^2$ about y-axis, between $x=1$ and $x=2$, using the shell method.
- Prove the trig identity $\sec x- \sin x \tan x =\frac{1}{\sec x}$
- Basic calc help
- Help doing a few trig proofs
- Easy money (basic calc)