piarepm

2021-12-30

Evaluate the integral ${\int}_{0}^{\frac{\pi}{2}}\frac{{\mathrm{sin}}^{3}x}{{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{3}x}dx$

sonorous9n

Beginner2021-12-31Added 34 answers

As

${\int}_{a}^{b}f\left(x\right)dx={\int}_{a}^{b}f(a+b-x)dx$ ,

If

$I={\int}_{0}^{\frac{\pi}{2}}\frac{{\mathrm{sin}}^{n}x}{{\mathrm{sin}}^{n}x+{\mathrm{cos}}^{n}x}dx$

$={\int}_{0}^{\frac{\pi}{2}}\frac{{\mathrm{sin}}^{n}(\frac{\pi}{2}-x)}{{\mathrm{sin}}^{n}(\frac{\pi}{2}-x)+{\mathrm{cos}}^{n}(\frac{\pi}{2}-x)}dx$

$={\int}_{0}^{\frac{\pi}{2}}\frac{{\mathrm{cos}}^{n}x}{{\mathrm{cos}}^{n}x+{\mathrm{sin}}^{n}x}dx$

$\Rightarrow I+I={\int}_{0}^{\frac{\pi}{2}}dx$

assuming${\mathrm{sin}}^{n}x+{\mathrm{cos}}^{n}x\ne 0$ which is true as $0\le x\le \frac{\pi}{2}$

Generalizition:

If$J={\int}_{a}^{b}\frac{g\left(x\right)}{g\left(x\right)+g(a+b-x)}dx,\text{}J={\int}_{a}^{b}\frac{g(a+b-x)}{g\left(x\right)+g(a+b-x)}dx$

$Right\to owJ+J={\int}_{a}^{b}dx$

provided$g\left(x\right)+g(a+b-x)\ne 0$

If$a=0,b=\frac{\pi}{2}$ and $g\left(x\right)=h\left(\mathrm{sin}x\right)$ ,

$g(\frac{\pi}{2}+0-x)=h\mathrm{sin}(\frac{\pi}{2}+0-x))=h\left(\mathrm{cos}x\right)$

So J becomes

If

assuming

Generalizition:

If

provided

If

So J becomes

Corgnatiui

Beginner2022-01-01Added 35 answers

Symmetry! This is the same as the integral with ${\mathrm{cos}}^{3}x$ on top.

If that is not obvious from the geometry, make the change of variable$u=\frac{\pi}{2}-x$

Add them, you get the integral of 1.. So our integral is$\frac{\pi}{4}$ .

If that is not obvious from the geometry, make the change of variable

Add them, you get the integral of 1.. So our integral is

Vasquez

Expert2022-01-09Added 669 answers

Hint: if

and

Then consider

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?