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2015年9月15日 · Integrate $$\int \tan^6x\sec^3x \ \mathrm{d}x$$ I tried to split integral to $$\tan^6x\sec^2x\sec x$$ but no luck for me.

Your substitution is not correct. You should still still have a $\sin^3 x$ upstairs. However when integrating a product of an even power of $\tan$ with an even power of $\sec$, you can do the following, which takes advantage of the facts that $\tan^2x+1=\sec^2 x$ and that the derivative of $\tan x$ is $\sec^2 x$:

2024年11月19日 · Using, as @AmrutAyan already did in comments, the Taylor series of $\tan(t)$, the problem is $$\int\tan(x)\tan(\tan (x))\, dx=\sum_{n=1}^\infty\frac{(-4)^n \left(1-4 ...

I am trying to find the integral of $$\int \tan x \sec^3 x dx$$ $$\int \tan x(1+\tan^2 x)\sec x\, dx$$ This gets me nowhere since I get a $\sec^2 x$ derivative with tan substitution so I try some...

How to integrate $\int\sqrt{\frac{1+x}{1+x^{2}}}dx$? Hot Network Questions What kind of scam is happening here, and is my next step to call my own bank or the institution that sent me money?

2015年3月4日 · Only by doing this in two harder ways did I catch a really nice trick for this problem in particular. By doing $\tan^2(x)=\sec^2(x)-1$ we get

2017年11月23日 · The integral of: $$\int\tan^n(x)dx$$ I found this proof online for the answer and it went like this: $$\int\tan^{n-2}(x)\tan^2(x)dx$$ $$\int\tan^{n-2}(x)(\sec^2(x)-1)dx$$ $$\int\tan^{n-2}(x)\sec^2(x)dx-\int\tan^{n-2}(x)dx$$ Then using u-substitution it yielded the reduction formula: $$\frac{\tan^{n-1}(x)}{n-1}-\int\tan^{n-2}dx$$

Here's my attempt at the problem: $\\int{\\tan^{5}(x)\\sec^4(x)}dx= \\int{\\frac{\\sin^5(x)}{\\cos^9(x)}}\\,dx= \\int{\\frac{\\sin(x)(\\sin^2(x))^2}{\\cos^9(x)}}\\,dx ...

I have a problem when trying to solve this question Question. What is the answer of the indefinite integral $$\\int x\\tan x \\; dx?$$ Maple gives a complicated answer based on the series. Is there ...