I have to solve this integral using Cauchy's integral formula. I tried to substitute it with several different attempts but without a solution. Can anyone help? $$\int_0^{2\pi} \frac{1}{a^2 \cdot {\cos}^2(t) + b^2 \cdot {\sin}^2(t)} dt$$
123378
Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b&gt;0$.   My Try:  So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq 2\pi\}$.   On $\Gamma$, $z=a\cos t+i b \sin t$, so $|z|^2=z\bar{z}=a^2\cos^2 t+b^2 \sin^2 t$.   Now, $dz=-a\sin t+i b \cos t dt$.   Hence, the integral becomes $\displaystyle \int_\Gamma \frac{dz}{z(iab+(\sin t \cos t)(b^2-a^2))}$. I know that $\displaystyle \int_\Gamma \frac{dz}{z}=2\pi i$. Now, how do I get rid of $\sin t \cos t$ part? I am stuck here. Can somebody please explain how?
30958
How do we integrate  $$\int \frac{x^2+20}{(x \sin x+5 \cos x)^2}dx$$  Could someone give me some hint for this question?
100511
I have a complicated integral to solve. I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts. are there any simpler methods to evaluate this integral?  $$ \int\!\sin (101x)\cdot\sin^{99}(x)\, dx $$
75595
I have been attempting to solve this integral for several days in preparation for an exam and keep reaching dead ends.  I(a,b) = $\int_0^{\pi/2} \ln(a^{2}\cos^2(x)+b^{2}\sin^2(x)) dx$, where $a,b&gt;0$  My best approach so far is defining:  $F(x) = I(x, b)$  Then, I know:  $F(b) = I(b, b) = \pi/2 * \ln(b^2)$  Then I am now attempting to find $F'(x)$ using differentiation under the integral sign, and integrating from b to a.  This is where I am stuck - no matter what I do, I cannot solve the integral. Am I in the right direction or should this be solved in a different manner? I also considered something elliptical due to the acos, bsin in the original integral, yet it has also reached a dead end.
47738
While solving another problem I have come across this integral which I am unable to evaluate. Can someone please evaluate the following integral? Thank you.      $$\int_0^{2\pi}\frac{1}{(2+\cos\theta)^2}\,d\theta.$$
29276
I need to find the cauchy principle value of $\int_{-\infty}^{\infty}\sin(x)/(x-a)dx$ ? I'm think of rewriting in terms of $e^{i\theta}$ and try to rewrite as contour integral?   Need some aid on how to calculate this integral please. Thank you for help
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.
19389
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it by methods of complex analysis. But I have not taken this course yet. Thanks for any help.
72470
I am trying to solve this integral, I think that it could be solve using the complex.   $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
4458
Find the integral  $$ \int_0^{\pi} \frac{x \sin x}{1+(\cos x)^2}$$
79510
I think this question would be with complicated methods in singularity or other kind of methods. I need to solve the integral $$\int_0^\infty\frac{\cos(ax)-\cos(bx)}{x}dx$$ Thanks a lot.
85817
How do I solve this? $$\int_0^\infty\frac{\ln(1+{x^2})}{1+{x^2}}$$ I know the answer is $\pi\ln2$.
92455
$$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx. $$  I am not sure how to solve this question. Can anyone help me to approach this problem. Thanks.
101742
I need solve this integral, and I tried various methods of solving and did not get it. The integral is:   $$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$$  where $t$ is a positive integer.
28624
Calculate the integral for $\left|a\right|&lt;1$  $$\int_{-\pi}^{\pi} \dfrac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$$  I'm supposed to evaluate this using method of residues, but the parameter a is confusing me. I'm not even sure what contour I should use. Any help would be very appreciated.
57753
I was asked to solve this indefinite integral using Integration by parts.  $$\int \sqrt{1-x^2} dx$$  I know how to solve if use the substitution $x=\sin(t)$ but I'm looking for the Integration by parts way.   any help would be very appreciated.
106957
$$\int_0^\pi x \ln(\sin (x))dx  $$   I tried integrating this by parts but I end up getting integral that doesn't converge, which is this $$ \int_0^\pi \dfrac{x^2\cos (x)}{\sin(x)} \ dx$$ So can anyone help me on this one?