Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of .

There are two main kinds of integrals:

  • definite integrals (e.g. proper and improper integrals), which often have numerical values
  • indefinite integrals, which group families of functions with the same derivative.

Several techniques to solve integrals have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

73636 questions
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How to compute this integral...

I need to compute $$ \int_0^1 \frac{\arctan x}{(1+x)^2}dx$$ I tried substituting $t= \arctan x$ (check please), getting $$ \int \frac{t}{1+2\sin t \cos t}dt $$ but I still can't handle it. Any help would be useful: I haven't done integrals for…
Francesco Gemma
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$\int\frac{a^2\sin^2x+b^2\cos^2x}{a^4\sin^2x+b^4\cos^2x}dx$

$\int\frac{a^2\sin^2x+b^2\cos^2x}{a^4\sin^2x+b^4\cos^2x}dx$ My…
Brahmagupta
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If $\int_a^b f(t) \, dt$ exists and is positive, then there is a subinterval $I$ of $[a,b]$ and $m>0$ such that $f(x) \geq m$ throughout $I$.

Proof: If $\int_a^b f(t) \, dt$ exists and is positive, then there is a subinterval $I$ of $[a,b]$ and a constant $m>0$ such that $f(x) \geq m$ throughout $I$. Hint: Consider $L(f,P)$
user289293
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Computing volume of $ D = \{ (x,y,z) \in \mathbb{R}^{3} : x^2 + y^2 + z^2 \leq 1, y^2 + z^2 \leq x^2 \} $

I want to compute the volume given by: $$ D = \{ (x,y,z) \in \mathbb{R}^{3} : x^2 + y^2 + z^2 \leq 1, y^2 + z^2 \leq x^2 \} $$ Which acording to Geogebra looks like: Plot of D (Sorry guys, I tried to add some fancy plot of $ D $, but it looks like I…
D'oh
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What does this set look like?

I have to find the volume integral of the set $$M=\{x \in \mathbb{R}^2 \mid 1 \leq x_1^2 + x_2^2, |x_1| \leq 1, |x_2| \leq 1\}$$ But I can't figure how it looks like, so I can't set the bounds of the integral. Does anyone know how it looks like in…
JavaForStarters
  • 157
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Evaluating $\int_0^{\sqrt{3}}{\frac{\sqrt{1+x^2}}{x}}\,dx$

Could someone please show me how to evaluate this integral (maybe doing all the steps)? $$\int_0^{\sqrt{3}}{\frac{\sqrt{1+x^2}}{x}}\,dx$$ I prefer if you avoid to follow the same method used by WolframAlpha (with $\csc$, $\sec$ ecc). This is what I…
Overflowh
  • 483
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How to integrate exponential and power function?

I am trying to solve the following integral $$\int_{0}^{\infty}e^{-(ax+bx^c)}\,dx ; ~~~a,b,c>0.$$ I tried using partial functions but that didn't lead to anything. Any suggestion?
ALPHA
  • 515
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Calculate $\lim\limits_{R\rightarrow\infty}\int_0^\pi \cos(R\cos t)dt$ w.o. Bessel function

Im calculating a complex path integral to calculate $\int_0^\infty \frac{\sin x}{x}dx$. I was able to evaluate everything except the arc $\int_0^\pi i~\exp(iR~e^{it})dt$ where $R$ is the radius. I managed to use the $\lim\limits_{R\rightarrow…
meneken17
  • 162
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Difficult Integral

I have a problem with solving this integral: $$\int{\frac{2x^2+3x+1}{\sqrt{x^2+1}}} dx$$ I tried to use substitution but I got stuck. Can anyone help me?
jane
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Integrate expression with $x^6$

So I'm trying to integrate this expression, but I'm not figuring out what's the best substitution to do... $ \int \frac {1}{x^6+1} dx $ I tried to take $x^6 +1 $ and write $ (x^2 + 1) (x^4 -x^2 + 1) $ and then do partial functions, so I reach to…
Granger Obliviate
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Iterated integral exists, but is not integrable

On the domain $D:[0,1]\times[0,1]\times[0,1], ~f(x,y,z)$ is defined to be: $$ \begin{align*} &1~ \text{for rational}~ x,\\ &2~ \text{for irrational}~ x,~0\leq y \leq \frac{1}{2}\\ &0~ \text{for irrational}~ x,~\frac{1}{2}< y \leq…
user245273
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Fourier integral

how could i evaluate the following integral ?? $$\int_{-\infty}^\infty dt \frac{\exp(-iut)}{|at|^{1/2+ib}}$$ here $a$ and $b$ are positive real numbers.. how can i make this integral ? thanks. $ |x| $ means the absolute value function I think this…
Jose Garcia
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Reverse the order of integration and evaluate the integral

Reverse the order of integration and evaluate the integral: $$\int_0^1 \int_x^\sqrt{x}{e^{x\over y}} \, dy \, dx$$ The answer is supposed to be $\frac{e}{2}-1$ but I keep getting $\frac{x}{2}(e-1)$. This is my work: $$\int_0^\sqrt{x}…
juliodesa
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Evaluation of $ \int_{-k}^{k}\frac{1}{\sqrt{\cos x-\cos k}}dx$

Evaluation of $$\displaystyle \int_{-k}^{k}\frac{1}{\sqrt{\cos x-\cos k}}dx\;,$$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int_{-k}^{k}\frac{1}{\sqrt{\cos x-\cos k}}dx = 2\int_{0}^{k}\frac{1}{\sqrt{\cos x-\cos k}}dx$$ Now Substiute $$\displaystyle…
juantheron
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Integration involving d[x] instead of dx

Find the value of: $$\int_0^3(x^2+1)d\lfloor x\rfloor$$ Shouldn't the answer be zero since $[x]$ is always an integer? But the options given are $12,17,15,19$. ([x] denotes floor(x)) From $0$ to $1$, $[x]=0$ and $d[x]=0$. Similarly, from $1$ to $2$,…
Aditya Dev
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