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I am trying to evaluate the integral $$\int \frac {1} {1+x^4} \mathrm dx.$$ The integrand $\frac {1} {1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I ...
Others answered about how cos(i) c o s (i) can be calculated using Euler's formula. But I will elaborate from a different perspective. We know that cosine function can be defined geometrically for certain real numbers and using further geometric arguments can be extended to entire real numbers. But how to extend it to complex plane? One of the natural ways to do is by defining
$$\\lim_{x\\to1}\\frac{m}{1-x^m} -\\frac{n}{1+x^n} \\;\\;\\;\\;\\;\\; m,n\\in \\mathbb{N}$$ My teacher had given the class this sum as homework. He gave us a hint ...
The following is a question from the Joint Entrance Examination (Main) from the 09 April 2024 evening shift: $$ \lim_ {x \to 0} \frac {e - (1 + 2x)^ {1/2x}} {x} $$ is equal to: (A) $0$ (B) $\frac {-2} {...
I am currently stuck on this question and need some help in figuring out where my mistake is. Take complex function $f(z) = \\frac{z^6}{(1 + z^4)^2}$ and integrate ...
Spherical Coordinate Homework Question Evaluate the triple integral of $f (x,y,z)=z (x^2+y^2+z^2)^ {−3/2}$ over the part of the ball $x^2+y^2+z^2\le 81$ defined by ...
I was playing around with double sums and encountered this problem: Evaluate $$\sum_ {i=1}^ {\infty} \sum_ {j=1}^ {\infty} \frac {1} {ij (i+j)^2}$$ It looks so simple I thought it must have been seen befo...
