Tagged Questions

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Integral of $\frac1{\cos^n x}$

Hi guys I have already proven for an assignment that: $$\int\cos(x)^n dx=\frac{1}{n}\cos(x)^{n−1}\sin(x) + \frac{n-1}{n}\int\cos(x)^{n−2}dx$$ Now we have been asked to calculate ...
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Real life math to explore/solve

What are some examples of mathematics application in the real life that is interesting to explore about? And not too complicated but not too easy, something that exist around us. I'm interested in ...
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Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$\displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x.$$ From this ...
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Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$  I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$  The confusing part is : What has ...
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Evaluate $\int\tan x\, dx$

$-1 = 0$ by integration by parts of $\tan(x)$ The solutions to the above problem state that you can't cancel the integrals on each side because they both have an unknown constant attached to them. ...
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How to integrate $\int\frac{t^2}{\sqrt{4t-t^2}} \, dt$ using trig substitution.

How do I integrate $\int\frac{t^2}{\sqrt{4t-t^2}} \,dt$? I solved this integral by a very long process(over 2 pages of work) and I got the answer of ...
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Algebraically, how are $-\ln|\csc x + \cot x| +C$ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C$ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
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Evaluating $\int \:\sqrt{1+e^x}dx$ , why I got different answers?

I've got 2 steps to evaluating $\int \:\sqrt{1+e^x}dx$ which lead to different values first step : $\int \:\sqrt{1+e^x}dx$ let $u\:=\:\sqrt{1+e^x}$ , $du\:=\:\frac{e^x}{2\sqrt{1+e^x}}dx$ , but ...
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Integrating $\int\frac{\sqrt{16x^2-9}}{x}dx$?

I am trying to differentiate from my previous question, but I am having trouble in the finishing steps. I have the integral $\int\frac{\sqrt{16x^2-9}}{x}dx$. $$v=4x \hspace{15pt}dv=4dx$$ ...
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$\int \dfrac{\cos x}{\left(\cos (2x)\right)^{3/2}} dx$

Wolfram gives this nice result: $$\int\frac{\cos x dx}{\cos^{3/2}2x}=\frac{\sin x}{\sqrt{\cos 2x}}+\text{constant}$$ I have tried writing $\cos 2x = \cos^2x - \sin^2x$ and doing Weierstrass ...
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Integrate a trigonometrical expression [closed]

I know that this may sound a silly question but is the following which is the integral: $\int(1+(\cot^2)x)dx$
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Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$
The following integral may seem easy to evaluate ... $$\int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right).$$ Could you prove ...