Solve Calculus Questions with Step-by-Step Solutions

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1. Do you know the taylor series of ln (1+x) ? That would help.

2. Come on mate you can do this one. Don't let the absolute value scare you, just replace it with its piece wise definition! If x > 0, |x| = x. If x < 0, |x| = -x and if x=0, |x| = 0.

Find the derivative for the two separate pieces. Find the correct derivatives (check your working again), then set them equal to 0 for the extreme points and solve.
 
so i make two separate equations and solve them
is there a chance that one of them will cancel out??
 
Not two separate equations, just a single function that defined in terms of TWO pieces.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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