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.

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Evaluate $\int_0^2\frac{x^5}{\sqrt{x^3+6}}\,dx.$

I am stuck on the following integral: $\displaystyle\int_0^2\dfrac{x^5}{\sqrt{x^3+6}}\,dx.$ I have no idea how one can work it out. Normally I'd try $u=x^3+6$ but this surely does not work here.
bibo_extreme
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integral resulting in Bessel function

Prove that $$\int_{0}^{\infty} \sin \left(x\right) \sin \left(\frac{a}{x}\right) \ dx = \frac{\pi \sqrt{a}}{2} J_{1} \left( 2 \sqrt{a} \right)$$ where $J_{1}$ is the Bessel function of the first kind of order 1. Some calculations I have…
wnvl
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Why is $\vec{s}=\frac{\vec{r}}{V^\frac{1}{3}} \Leftrightarrow d\vec{s}=\frac{d\vec{r}}{V}$?

I am following a course which contains a part in statistical thermodynamics. One of the questions involves the partition function $Q_N$. I could not figure out the answer of the question myself, so I had a look in the solutions provided by the…
Stijn
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meaning of integration

I read that integration is the opposite of differentiation AND at the same time is a summation process to find the area under a curve. But I can't understand how these things combine together and actually an integral can be the same time those two…
trig
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How to integrate this function with $\ln(x)$ in the numerator?

I have to integrate $$ \iint {\log \sqrt x \over xy} \, \mathrm dx \, \mathrm dy$$ I don't really care about integrating over $y$, I can't begin to integrate over $x$ because I don't know how. What rules do I use to do this, I'm a bit rusty.
tokola
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Integrate without substitution

I'm wondering how I could integrate the following without substitution:$$\int \frac{4}{1 + e^{-x}}dx$$ I know we can factor out the constant so that $4 * \int \frac{1}{1 + e^{-x}}$ but I'm stumped as of what to do next. Could anyone help me out?
user2451412
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When to use Integral Substitution?

$e^x$$(1+e^x)^{1\over{2}}$ why can't use integration by part, What is meant by in the form of f(g(x))g'(x)? Can you give a few example? Thank you
problematic
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Double integral polar coordinates for area bounded by curves

I need to find the area bounded by the curves : $$ x^2 + y^2 = 1, \ y^2= x\sqrt3, \ x \geqslant \frac{y^2}{\sqrt3} $$ My attempt: $$ \int^{\frac{\pi}{2}}_{\frac{3\pi}{2}} \int^{\text{some complicated number}}_{-\text{some complicated number}}…
user3528592
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Evaluate $\int\sec^4(u) \operatorname d \!u$

Evaluate $$\int\sec^4(u) \operatorname d \!u$$ I don't know what to substitute: I've tried $1+\tan(u)$ and integration by parts. I know the general formula for $\sec^n(u)$, but I want to be able to do this integral on my own.
ahorn
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solve indefinite integral

I have this indefinite integral $\int 3 \sqrt{x}\,dx$ to solve. My attempt: $$\int 3 \sqrt{x}\,dx = 3 \cdot \frac {x^{\frac {1}{2} + \frac {2}{2}}}{\frac {1}{2} + \frac {2}{2}}$$ $$\int 3 \sqrt{x}\,dx = 3 \frac{x^{\frac {3}{2}}}{\frac {3}{2}} =…
S4M1R
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Integration of $4xe^{-1/2}$

It is given that $$\int_0^p4xe^{-\frac{1}{2}x}dx=9$$ where $p$ is a positive constant (i) Show that $$p=2 \ln \left( \frac{8p+16}{7} \right )$$ I reached $$8pe^{-p/2} + 16e^{-p/2} = 7$$ What are the steps to show in (i) ??
Arodi007
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Integrate $\csc(x)^3dx $

I'm not sure trig identity to use. Would we use $1+\cot^2(x) = \csc^2(x)$? I know that we break down into $\csc^2(x)$ and $\csc(x)$. Could I get hints?
user159778
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How to find $\left\lfloor\sum_{r=1}^{80}\int_0^1x^{\sqrt r-1}dx\right\rfloor$

$$\left\lfloor\sum_{r=1}^{80}\int_0^1x^{\sqrt r-1}dx\right\rfloor$$ My try: $$K=\left\lfloor\sum_{r=1}^{80}\int_0^1x^{\sqrt r-1}dx\right\rfloor=\left\lfloor\sum_{r=1}^{80}\frac1{\sqrt r}\right\rfloor=\left\lfloor\frac1{\sqrt 1}+\frac1{\sqrt…
RE60K
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Having trouble with simple integral

sorry I'm having some trouble evaluating this integral $\frac{dv}{dt} = -k(v-gt)^2-g$ where g and k are constants I'm assuming you just separate and integrate but I cannot seem to get it to work out.
fred
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Integration by parts with c value?

Ive asked this question a few times but still don't understand how to go about and there have been a few answers that a different so can anyone clarify what the correct answer for this integral would be - including the $c$ value? Calculate…
ojando
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