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
- 2122 views
- $20.00
Related Questions
- Show that the MLE for $\sum_{i=1}^{n}\left(\ln{2x_i} - 2\ln{\lambda} - \left(\frac{x_i}{\lambda}\right)^2\right)$ is $\hat{\lambda} = \sqrt{\sum_{i=1}^{n}\frac{x_i^2}{n}}$.
- Find and simplify quotient
- Equations of Motion and Partial Fractions
- Get the volume and surface area of the paraboloid $z=4-x^2-y^2$ cut by the plane $z=4-2x$
- Derivatives again. Thank you!
- Integration and Accumulation of Change
- Evaluate $\int \ln(\sqrt{x+1}+\sqrt{x}) dx$
- Find the antiderrivative of $\int \frac{v^2-v_o^2}{2\frac{K_e\frac{q_1q_2}{r^2}}{m} } dr$