# Questions tagged [indefinite-integrals]

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

3,693 questions
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### Is there a rule of integration that corresponds to the quotient rule?

When teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivatives ...
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### Integration - Primitives - Antiderivatives

Please help to calculate: $$\int\sqrt {{r}^{2}-{x}^{2}}{dx},\quad x\in[0,r]$$ Do any method of trigonometric substitution? Thanks.
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### What technique would be suitable to solve this: $\int \sin ^{5}\left( x^{2}\right) \left( x\cos \left(x^{2}\right)\right)\mathrm{d}x$

I think integration by parts might work but I'm now sure. Thanks very much.
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### How to calculate $\int \sqrt{(\cos{x})^2-a^2} \, dx$

How to calculate: $$\int \sqrt{(\cos{x})^2-a^2} \, dx$$
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### Integral with a substitution

I must calculate a following integral $$\int \frac{dx}{x^{2}\sqrt{1+x^{2}}}$$ with a subsitution like this $x = \frac{1}{t}, t<0$ I'm on this step $$\int \frac{dt}{\frac{1}{t}\sqrt{t^{2} + 1}}$$ ...
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### Not sure how to go about solving this integral

$\displaystyle \int \left( \frac{1}{x^2+3} \right)\; dx$ I've let $u=x^2+3$ but can't seem to get the right answer. Really not sure what to do.
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### Integral of floor function: $\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$

How would you go about solving integral of a floor? The particular problem I have is: $$\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$$
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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 ...
### Integrating $\int\frac{2dx}{x\ln(6x)}$
I needed some help integrating this: $$\int\frac{2\,dx}{x\ln(6x)}.$$ I have never seen the dx within the problem like that, I am assuming I can't just move it to the outside can I? Can I start by ...
Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx$ I know that the correct answer can be obtained by doing: $\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}... 11answers 8k views ### Ways to evaluate$\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}$and then ... 4answers 10k views ### Integral of$\sqrt{1 + \sqrt{x}}\$
My professor wants us to do this problem to refresh ourselves with substitution. We have to solve: $$\int\sqrt{1 + \sqrt{x}}\,\mathrm dx$$ $$\int\sqrt{1 + \sqrt{1 + \sqrt{x}}}\,\mathrm dx$$ ......