Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$  How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).  Any guidelines please?
1542
Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$   How can I get this formula by hand? Is there any simpler idea than using $u = \sin t$?     Is there a simple way to calculate $$ \int_0^{\pi/2}\frac{\sin^n t}{\sin^n t+\cos^n t}\,\mathrm dt $$ for $n&gt;3$? Could anyone come up with a reference for this exercise?
95678
could you help me on this integral ? $$\int \frac{1}{\cos(x)}\,\mathrm dx$$ Here's what I've started : $$\int \frac{1}{\cos(x)}\,\mathrm dx = \int \frac{\cos(x)}{\cos(x)^2}\,\mathrm dx = \int \frac{\cos(x)}{1-\sin(x)^2}\,\mathrm dx$$ Now, I did : $u = \sin(x)$, so $\mathrm du = 1$. Now I have : $$\int \frac{\text{???}}{1-u^2}\,\mathrm du$$ But at this point, I think I did the most of the job but I'm stuck. Could you help me to solve this integral please (to the integration by substitution at the end) ? Thanks EDIT : Now I follow the steps and I got : $$\int \frac{1}{1-u^2}\,\mathrm du$$ Doing the partial fraction I got $A = 1/2$ and $B = 1/2$. So basically I have : \begin{align} &amp; \int \frac{1}{1-u^2}\,\mathrm du = \int \frac{1/2}{1+u}\,\mathrm du + \int \frac{1/2}{1-u}\,\mathrm du \\[8pt] = {} &amp; \frac 1 2 \left(\int \frac{1}{1+u} \, du - \int \frac{1}{1-u} \, du\right) \\[8pt] = {} &amp; \frac 1 2 \ln\left(\frac{1+u}{1-u}\right) = \ln\left(\left(\frac{1+\sin(x)}{1-\sin(x)}\right)^{1/2}\right) \\[8pt] = {} &amp; \ln \left(\frac{\sqrt{1+\sin(x)}}{\sqrt{1-\sin(x)}}\right) = \ln\left(\frac{\sqrt{1+\sin(x)}}{\sqrt{\cos(x)^2}}\right) \\[8pt] = {} &amp; \ln\left(\frac{\sqrt{1+\sin(x)}}{\cos(x)}\right) \end{align} Here's what my teacher got :  What's wrong with what I did ? Did I miss something ?
96009
$$\int \frac{\mathrm{d}x}{\sqrt{x}+\sqrt{x+1}+\sqrt{x+2}}$$  I tried substituting $x=z^2$ ... also $x=\tan^2 \theta$ ... but couldn't solve it either ways... if someone can help then it would be good.
97632
Find the value of $$\int\frac{1}{(x^2+x+1)^{1/2}} dx$$   Anyone can provide hint on how to integrate this, and how you know what method to use? (I mean, is there any general guideline to follow for solving?)  Thank you!
8650
I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}\mathrm dx$. Do I have to use trigonometric substitution?
23127
Calculate the definite integral      $$ I=\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\;\mathrm dx $$      given that $a&gt;b&gt;0$.   My Attempt:  If we replace $x$ by $C$, then  $$ I = \int_{0}^{\pi}\frac{\sin^2 C}{a^2+b^2-2ab\cos C}\;\mathrm dC $$  Now we can use the Cosine Formula ($A+B+C=\pi$). Applying the formula gives  $$ \begin{align} \cos C &amp;= \frac{a^2+b^2-c^2}{2ab}\\ a^2+b^2-2ab\cos C &amp;= c^2 \end{align} $$  From here we can use the formula $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$ to transform the integral to  $$ \begin{align} I &amp;= \int_{0}^{\pi}\frac{\sin^2 C}{c^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2A}{a^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2 B}{b^2}\;\mathrm dC \end{align} $$  Is my process right? If not, how can I calculate the above integral?
98316
I am trying to show  $$\int_{0}^{\infty}\frac{\sin^{3}(x)}{x^{3}}\mathrm dx = \frac{3\pi}{8}.$$  I believe the contour I should use is a semicircle in the upper half plane with a slight bump at the origin, so I miss the singularity.  Lastly I have the hint to consider  $$\int_{0}^{\infty}\frac{e^{3iz}-3e^{iz}+2}{z^{3}}\mathrm dz$$  around the contour I mentioned. Thanks for any help or hints!
44141
How do I evaluate $$\int \frac{\sec^2 x}{(\sec x + \tan x )^{{9}/{2}}}\,\mathrm dx$$  I've started doing this problem by taking $u=\tan x$.
102169
Evaluating $$\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$$   I had read the , but it doesn't work.
52543
I have no idea how to start, it looks like integration by parts won't work.  $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$  If someone could shed some light on this I'd be very thankful.
108659
How do I compute the integration for $a&gt;0$,   $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$  I want to find a complex function and integrate by the .
10157
I have stumbled onto an interesting integral$$\int_0^\infty \sin(x)\sin\left(\frac{1}{x}\right)dx$$ which I noticed graphically that it appears to be $1$, but I have no idea on how to evaluate it. Maybe it could be done with the use of Bessel Functions? Any help is appreciated.
87616
In evaluating integral $\int\sin^{3}x dx $ I am pretty sure we need to use substitution $e^x=t$, but can't go  next step.
85114
I'm having trouble computing the integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$ I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as $u=\sin(x)$ and $u=\cos(x)$, but it was not very effective.  Any suggestions are welcome. Thanks.
17869
$$\int \frac{\sin\left(x\right)}{1+x^2}dx$$  I have tried to integrate by parts but it doesn't work. How do I evaluate it? Any advice, hint or well-thought solution will be appreciated.
3572
The question:      Compute$$ \int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2} $$   Tried integration by parts. That didn't work.  How do I proceed?
39298
I'd love your help with finding the sign of the following integral: $\int_{0}^{2 \pi}\frac{\sin x}{x} dx$,I know that computing it is impossible.  I tried to use integration by parts and maybe to learn about the sign of each part and conclude something but It didn't work for me.  Any suggestions?