Michael Maggard

2022-01-06

Evaluate the following integral:
$\int \frac{\sqrt{\mathrm{sin}x}}{\sqrt{\mathrm{sin}x}+\sqrt{\mathrm{cos}x}}dx$

Fasaniu

$I=\int \frac{\sqrt{\mathrm{sin}x}}{\sqrt{\mathrm{sin}x}+\sqrt{\mathrm{cos}x}}dx=\int \frac{\sqrt{\mathrm{tan}x}}{1+\sqrt{\mathrm{tan}x}}dx$
substitute $\mathrm{tan}x={t}^{2}$ and $dx=\frac{1}{1+{t}^{4}}dt$
$I=\int \frac{t}{\left(1+t\right)\left(1+{t}^{4}\right)}dt=\frac{1}{2}\int \frac{\left(\left(1+{t}^{4}\right)+\left(1-{t}^{4}\right)\right)t}{\left(1+t\right)\left(1+{t}^{4}\right)}dt$
$=\frac{1}{2}\int \frac{t}{1+t}dt+\frac{1}{2}\int \frac{\left(t-{t}^{2}\right)\left(1+{t}^{2}\right)}{1+{t}^{4}}dt$
$=\frac{1}{2}\int \frac{t}{1+t}dt+\frac{1}{2}\int \frac{\left(t-{t}^{2}\right)\left(1+{t}^{2}\right)}{1+{t}^{4}}dt$
$=\frac{1}{2}\int \frac{\left(1+t\right)-1}{1+t}dt+\frac{1}{2}\int \frac{t+{t}^{3}-\left({t}^{2}-1\right)-{t}^{4}-1}{1+{t}^{4}}dt$
$=-\frac{t}{2}+\frac{1}{2}\mathrm{ln}|t+1|+\frac{1}{4}\int \frac{2t}{1+{t}^{4}}+\frac{1}{2}\int \frac{{t}^{3}}{1+{t}^{4}}dt-\frac{1}{2}\int \frac{{t}^{2}-1}{1+{t}^{4}}dt-\frac{1}{2}+C$
All integrals are easy expect

usaho4w

We rationalise the denominator to get
$I=\int \frac{\mathrm{sin}x-\sqrt{\mathrm{cos}x\mathrm{sin}x}}{\mathrm{sin}x-\mathrm{cos}x}dx$
Writing everything in terms of $\mathrm{cot}x$, we get
$I=\int {\mathrm{csc}}^{2}x\left(\frac{\sqrt{\mathrm{cot}x}-1}{{\mathrm{cot}}^{3}x-{\mathrm{cot}}^{2}x+\mathrm{cot}x-1}\right)dx$
Now substituting $u=\mathrm{cot}x$ and further $v=\sqrt{u}$ gives us
$I=-\int \frac{2v}{{v}^{5}+{v}^{4}+v+1}dv$
Performing a partial fraction decomposition we have
$I=\frac{2}{4-4\sqrt{2}}\int \frac{v+\sqrt{2}-1}{{v}^{2}+\sqrt{2}v+1}dv+\frac{2}{4+4\sqrt{2}}$
$\int \frac{v-\sqrt{2}-1}{{v}^{2}-\sqrt{2}v+1}dv+\int \frac{1}{1+v}dv={I}_{1}+{I}_{2}+{I}_{3}$
Hope you can take it from here

star233

HINT multiply nominator and denominator by $\frac{1}{\sqrt{\mathrm{sin}\left(x\right)}}$, then $t=\sqrt{\mathrm{cot}\left(x\right)}$ after all you'll have $\frac{2t}{\left({t}^{4}+1\right)\left(t+1\right)}$