# Questions tagged [indefinite-integrals]

Question about finding the primitives of a given function, whether or not elementary.

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### Mathematical Meaning of Antiderivatives

I'm largely a self-taught highschooler in basic Calculus and I'm utterly confused regarding what Indefinite integrals (or antiderivatives) do mean geometrically (if they really do), physically or ...
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### Solve the IVP $y'+2y = \frac{1}{1+x^2}$

Find the solution of the DE $$y'+2y = \frac{1}{1+x^2}\,\,\,\,\,\,\forall x \in \mathbb R$$ satisfying $y(0) = a$ where $a \in \mathbb R$ is a constant. My attempt: Since it's a linear ODE, therefore ...
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### How to integrate $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$?

Q) How to Integrate $\int \frac{3x^{4}+5x^{3}+7x^{2}+2x+3}{(x-6)^{5}}dx$ ? First of all let me tell what I think about this question. In my Coaching Institute, the chapter 'Integration' is over. This ...
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### $\int \frac{\sqrt{a^2-x^2}}{x^2}dx$ using trigonometric substitition

I'm aware this question is more easily done using substitution by parts or euler substitution, but this was under a section in my book where we were asked to use trigonemtric substations. Substituting ...
1 vote
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### Textbook says to integrate a fraction using 'Taylor's formula'?

I don't understand the solution my textbook gives for this problem: $$\int \! \frac{x^3}{(x+1)^5} \, \mathrm{d}x$$ I thought it had to be done with partial fractions, but I couldn't get it right, ...
1 vote
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### Why do we call integration "accumulation of change"?

So in virtually all English-language calculus classes I have seen, we define integration as the "accumulation of change". And that makes sense to me intuitively, but when I think about it, I ...
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### Evaluating the integral as a function of a matrix and a vector

Evaluate the integral $$\int{e^{{-1 \over 2}x^T{Yx+z^Tx}}dx}$$ as w.r.t $Y$ (matrix) and $z$ (vector) My attempt I have never seen such a problem before, and I have very unsure how to start. Since I'...
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### integral of $\int \frac {dx}{2\sin(x)^2 + 3\cos(x)^2}$

The answer is supposedly $$\frac {1}{\sqrt{6}} \arctan\left(\sqrt{\frac {2}{3}}\tan x\right) + C$$ So I need to get it into form $$\int \frac{\mathrm dx}{a^2+x^2}$$ but I am not sure what identities ...
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