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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How to integrate $\int\frac{\ln x\,dx}{x^2+2x+4}$

$$K=\int\frac{\ln x\,dx}{x^2+2x+4}$$ I did this $x^2+2x+4=(x+\alpha)(x+\beta)$, then used partial fraction, I am then unsure how to integrate $\int\frac{\ln x}{x+c}\,dx$. I tried Integration by parts also taking first function as both of them…
RE60K
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Integral of Derivative squared

For a function $f$ I know that: $$\int{f'(r)dr}=f(r)$$ where $f(r)$ is known. knowing the result of this integral how can i calculate $$\int{(f'(r))^2dr}$$ Is there any relation between these integrals?
Pouya
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Determine the value of the integral $I=\int_{0}^{1}\frac{\ln\left(1-a^2x^2\right)}{\sqrt{1-x^2}}dx$

Determine the value of the integral $$I(a)=\int_{0}^{1}\frac{\ln\left(1-a^2x^2\right)}{\sqrt{1-x^2}}dx, \: |a|\leq 1$$ My try: $\to I'(a)=\int_{0}^{1}\frac{-2ax^2}{\left(1-a^2x^2\right)\sqrt{1-x^2}}dx$ Set $x=\cos t\to dx=-\sin tdt$ Hence…
Iloveyou
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Evaluate $\int_0^e{W(x)}\,\mathrm{d}x$

The function $W(x)$ satisfies $W(x)e^{W(x)}=x$ for all $x$. Evaluate $$\int_0^e{W(x)}\,\mathrm{d}x$$ I tried integrating $xe^{-W(x)}$ but can't see how to do it.
user1510374
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How find this $\int_{0}^{\pi}\frac{\cos{(nx)}}{\cos{x}+a}dx$

Fin the integral $$I_{n}=\int_{0}^{\pi}\dfrac{\cos{(nx)}}{\cos{x}+a}dx$$ where $n\in {\mathbb N}\,,\ a>1$ My try: let $$I_{n}-I_{n-1}=\int_{0}^{\pi}\dfrac{\cos{(nx)}-\cos{(n-1)x}}{\cos{x}+a}dx$$ and…
user94270
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How to find this integral $I=\int_{-\pi}^{+\pi}\frac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx$?

Find the integral $$I=\int_{-\pi}^{+\pi}\dfrac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx$$ My try: let $$I=\int_{-\pi}^{0}\dfrac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx+\int_{0}^{+\pi}\dfrac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx=I_{1}+I_{2}$$ for $I_{2}$,we…
user94270
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For what functions does $\int\limits_{-\infty}^{\infty}x \sin(f(x))\,dx$ converge?

I'm thinking that the class of functions which cause the integral to converge are those whose big O is above $n^3$. Its not hard to show that if $f(x)$ is $x^3$, $x^4$, etc then the integral converges. I am having trouble with the function…
Searke
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Integral $ \int_0^{\pi/2} x \sqrt{\cot x} \, dx $

I cannot evaluate the following integral $$ \int_0^{\pi/2} x \sqrt{\cot x} \, dx $$ which means $$ \int_0^{\pi/2} \frac{x}{\sqrt{\tan x}} \, dx = 0.97482\ldots. $$ This integral seems to be equal to the value of $$ \frac\pi{2\sqrt2} \left( \frac\pi2…
Kei Tojo
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how to do integration $\int_{-\infty}^{+\infty}\exp(-x^n)\,\mathrm{d}x$?

how to do integration $\int_{-\infty}^{+\infty}\exp(-x^n)\,\mathrm{d}x$, assuming $n>1$ ? From wiki page Gaussian Integral: $\int_{-\infty}^{+\infty}\exp(-x^2)\,\mathrm{d}x = \sqrt{\pi}$ So, one can define a random variable $X$ has $\text{pdf}(x) =…
athos
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How to evaluate $\int\frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}\,dx$

How to evaluate the following pseudo-elliptic integral? $$\int\frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}\,dx$$ I think that I should make $(x+1)^4$ and substract something under the square root. I got $(x+1)^4 -x^4-3x^3-5x^2-3x-1$ but I have no idea what I…
James
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In which senses can an integral exist?

I asked about the value of an integral here: Hard integral that standard CAS get totally wrong The question got downvoted and voted to close because I didn't understand (and wasn't able to answer) the following question: In which sense is the…
vonjd
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Help for evaluating complicated integral $\int \frac 1 {x^n-x} dx$

I have this complicated integral to evaluate : $$\int \dfrac 1 {x^n-x} dx$$ I'm struggling to evaluate this. My attempt : $$\int \dfrac1x \cdot \dfrac 1 {x^{n-1}-1} dx$$ Now, I try to apply integration by parts. For that, I use : $V=\large\dfrac1x$…
anon1456
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Integral from Ramanujan to Hardy

I came across the interesting family of integrals that was studied by Ramanujan and was one of subjects of his first letter to Hardy. $$ \phi(n):=\int_{0}^{\infty} \frac{\cos n x}{e^{2 \pi \sqrt{x}}-1} d x $$ He offers a functional…
Ricardo770
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Simple u-subsitution - Paradoxical Result

If I were to try and take $$\int{\mathrm{sin}(t)\mathrm{cos}(t)dt} $$ I would either take $u=\mathrm{sin}(t) $, yeilding a result of $\frac{1}{2} \mathrm{sin}^2(t) + C$, or I would take $u=\mathrm{cos}(t) $, yeilding a result of $-\frac{1}{2}…
user1833028
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If $f(x)$ is integrable can we say that $f(x)^n$ is integrable?

Suppose $f(x)$ is a positive and continuously differentiable functions. In addition, it is well-known that $\int_{0}^{\infty} f(x)dx$ is bounded. My point of view is that $\int_{0}^{\infty} f(x)^mdx$ (where $m \in \mathbb N$) is bounded. I will…
majid
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