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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Does this integral expression makes sense?

I'm wondering whether this expression has some significance: $$\int_{-1}^1 \frac{dx}{\sqrt{|x|}}$$ And, in general, if expressions in the following form make sense: $$\int_a^b f(x) dx$$ Where the set $(a, b)$ contains points out of the domain of…
hey hey
  • 705
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$\int_0^\frac{\pi}{2}\frac{\ln(\sin(x))\ln(\cos(x))}{\tan(x)}dx$

I have the problem below: $$\int_0^\frac{\pi}{2}\frac{\ln(\sin(x))\ln(\cos(x))}{\tan(x)}dx$$ I have tried $u=\ln(\sin(x))$ so $dx=\tan(x)du$ so the integral becomes: $$\int_{-\infty}^0u\ln(\cos(x))du$$ but I cannot find a simple way of getting rid…
Henry Lee
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Find this integral $I=\int\frac{x}{(1-3x)^{3/2}(x+1)^{3/2}}dx$

Find the integral $$I=\int\dfrac{x}{(1-3x)^{3/2}(x+1)^{3/2}}dx$$ My try: $$(1-3x)(x+1)=-3x^2-2x+1=-3(x+1/3)^2+\dfrac{4}{3}$$ Thus $$I=\int\dfrac{x}{(\frac{4}{3}-3(x+\frac{1}{3})^2)^{3/2}}dx$$
math110
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Find $\int\sqrt{\sin x+\cos x}\,dx $

I am trying to solve the question $$ \int\sqrt{\sin x+\cos x}\,dx $$ Is their any substitution by which I can get the answer. I tried different substitution like i multiplied both numerator and denominator by $\sqrt{\sin x+\cos x} $ and uses sinx…
Samar Imam Zaidi
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What can be considered as a constant in integration?

Suppose I have an integral to evaluate, which is of the form , $$\int_0^x xf(t)\,dt$$ So , is it correct to consider the $x$ in the integrand as a constant and remove it outside the integral? I think the answer is yes because it does not change with…
Aditi
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about the gaussian integral $\int_0^{\infty} e^{-x^2}\ \mathrm{d}x$

I have an exercise which seem to be a method of calculating the Gaussian integral : Let $\displaystyle f(x)=\int_0^1 \frac{\exp(-x^2(t^2+1))}{t^2+1}\ \mathrm{d}t$ Study the deriviability of $f$, then conclude that $\displaystyle \int_0^{\infty}…
Tulip
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Calculating a difficult integral

For $p>0$, let $g(x)=\begin{cases} p\left[\dfrac{x}{p}\right]+\dfrac{p}{2}& x\ge 0,\\ -g(-x)& x<0 \end{cases}$ Try to prove that for all $x \in…
math110
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Is this really a double integral problem?

I'm solving a list of exercises of double integrals and they normally have a range for $x$ and $y$, but in this case it says that $y = x^2$ and $y = 4$, so I thought that $x$ would be $\sqrt{y}$, but my answer was wrong. The answer should be…
juliano.net
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Integral $\int_0^1 \frac{\sqrt x \ln x} {x^2 - x+1}dx$

I am trying to evaluate $$I=\int_0^1 \frac{\sqrt x \ln x} {x^2 - x+1}dx=\int_0^1 \frac{\sqrt x (1+x)\ln x} {1+x^3}dx$$ Now if we expand into geometric series: $$I=\sum_{n=0}^{\infty} (-1)^n \int_0^1 (x^{3/2}+x^{1 /2})x^{3n}\ln x dx$$ Also since…
Zacky
  • 27,674
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Help me to show $\int_1^\infty \frac{1}{x+x^3}dx = \frac{\ln 2}{2}$?

I have this exercise, and I get the right result. But while I think the first part is ok, the second part is from a formal stand point pretty hairy. So here the first part (I left some step out, but they should be for the most of you…
leo
  • 653
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Evaluate the integral$ \int_{-\infty}^{\infty}\frac{b\tan^{-1}\big(\frac{\sqrt{x^2+a^2}}{b}\big)}{(x^2+b^2)(\sqrt{x^2+a^2})}\,dx$.

I am attempting to evaluate $$\int_{-\infty}^{\infty}\dfrac{b\tan^{-1}\Big(\dfrac{\sqrt{x^2+a^2}}{b}\Big)}{(x^2+b^2)(\sqrt{x^2+a^2})}\,dx. $$ I have tried using the residue formula to calculate the residues at $\pm ib,\pm ia,$ but it got messy very…
JohnColtraneisJC
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Integrate $\int \ln(x^2 +1)\ dx$

$$\int \ln(x^2 +1)\ dx$$ I done it using integration by parts where $\int u\ dv = uv - \int v\ du$ Let $u$ = $\ln(x^2 +1)$ $du = \frac{2x}{x^2+1} dx $ Let $dv = dx$ so $v=x$ $\int \ln(x^2 +1)\ dx = x \ln (x^2 +1) - \int \frac{2x^2}{x^2+1} $ I…
user59439
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Integrating $\sqrt{\tan x}$

I attempted to evaluate $$\int\sqrt{\tan(x)} dx$$ but, according to wolfram, I must've made a mistake somewhere. I couldn't find it myself so could you tell me where I messed up ? $$\int \sqrt{\tan(x)}dx$$ $$u=\sqrt{\tan(x)}\qquad du =…
Adam
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Integrating a multiplication of functions where the first has a high exponent

Let's take this as an example. $$\int (x^{15}\ln x)dx $$ Is there a way to solve it someway clever? Using integration by parts 15 times would be wearisome...
0x6B6F77616C74
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Necessary condition for the integral equation $u(x) = f(x)+\lambda\int K(x,t)u(t)dt$ to have a continuous solution?

Is the following condition necessary for the integral equation $$u(x) = f(x)+\lambda\int K(x,t)u(t)dt$$ to have a continuous solution: $f(x) \neq 0$, is real and continuous in the interval $[a,b]$? When $f(x) = 0$, the integral equation will become…
Alka Goyal
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