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Questions tagged [indefinite-integrals]

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

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Indefinite integral of secant cubed

I need to calculate the following indefinite integral: $$I=\int \frac{1}{\cos^3(x)}dx$$ I know what the result is (from Mathematica): $$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$ but I don't ...
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Striking applications of integration by parts

What are your favorite applications of integration by parts? (The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!) Thanks for your ...
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Ways to evaluate $\int \sec \theta \, \mathrm d \theta$

The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then ...
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Evaluating the indefinite integral $\int \sqrt{\tan x} ~ \mathrm{d}{x}.$

I have been having extreme difficulties with this integral. I would appreciate any and all help. $$\int \sqrt{\tan x} ~ \mathrm{d}{x}.$$
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Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$

I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$ This problem is a classic, but I seem to be missing one step or the understanding of ...
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Integration of $\int\frac{1}{x^{4}+1}\mathrm dx$ [closed]

I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}\mathrm dx$. Do I have to use trigonometric substitution?
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Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$

I'm having trouble computing the integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$ I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as ...
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Integral of $\sqrt{1-x^2}$ using integration by parts

I was asked to solve this indefinite integral using Integration by parts. $$\int \sqrt{1-x^2} dx$$ I know how to solve if use the substitution $x=\sin(t)$ but I'm looking for the Integration by ...
525 views

Evaluating this integral $\small\int \frac {x^2 dx} {(x\sin x+\cos x)^2}$

The question: Compute$$\int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2}$$ Tried integration by parts. That didn't work. How do I proceed?
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What is the integration of $\int 1/(x^{2n} +1)dx$?

I am a student who is preparing for IIT exam. I was just practicing calculus and encountered this problem. I tried different substitutions but none of them seemed to work. So what is the integration ...
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Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
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$-1 = 0$ by integration by parts of $\tan(x)$

I had a calculus final yesterday, and in a question we had to find a primitive of $\tan(x)$ in order to solve a differential equation. A friend of mine forgot that such a primitive could easily be ...
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Solving this integral $\int\frac{1}{1+x^n} dx$?

Well, this might be a really simple one. But still... What will be the soln. to --- \begin{aligned} \int\frac{1}{1+x^n} dx \end{aligned} Is substituting \begin{aligned} 1+x^n \end{aligned} by tan z ...
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Demystify integration of $\int \frac{1}{x} \mathrm dx$

I've learned in my analysis class, that $$\int \frac{1}{x} \mathrm dx = \ln(x).$$ I can live with that, and it's what I use when solving equations like that. But how can I solve this, without ...
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What is the antiderivative of $e^{-x^2}$

I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$ So, I ...
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When the integral of products is the product of integrals.

I'm self-studying and was doing the following integral: $$I = \int \frac{e^{\frac{1}{x}+\tan^{-1}x}}{x^2+x^4} dx$$ I solved it fine by letting $u = \frac{1}{x} + \tan^{-1}x$. My question is ...
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Evaluate $\int \cos(\cos x)~dx$

Evaluate $\int \cos(\cos x)~dx$ I tried to use chain rule but failed. Can anyone help me please?
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Integral $\int\!\sqrt{\cot x}\,dx$

Find the integral $$\int\!\sqrt{\cot x}\,dx$$ How can one solve this using substitution? Can this be solved by complex methods?
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Families of functions closed under integration

What are some concrete families $\mathcal F$ of real functions that are closed under integration in the sense that for every $f \in \mathcal F$ there is $F \in \mathcal F$ such that $F'=f$? Here are ...
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Need help solving - $\int (\sin 101x) \cdot\sin^{99}x\,dx$

I have a complicated integral to solve. I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts. are there any simpler methods to evaluate this ...