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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Proof that the Integral of a Positive Function is Positive?

I've seen a few other posts about the integral of a positive function, it seems to hinge on it being discontinuous almost nowhere. So what's an example of a discontinuous almost everywhere function that is integrable, positive, and has a zero…
user82004
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Integrating a top heavy function

How do you integrate this function? $$\int\frac{x^3}{(x+5)^2}dx$$ I have tried it myself by substitution but I can't seem to get rid of the $x$s.
user124203
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Evaluating the integral $\int\frac{3x^{2}-x+2}{x-1}\;dx$

As the title suggests, the following integral has been given to me $$\int\frac{3x^{2}-x+2}{x-1}\;dx$$ Yet I still get the wrong answer every time. Can someone calculate it step-by-step so I can compare it to my own answer?
Kevin Jones
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Tricky looking integration (after separation of variables)?

I've come across something in my notes that jumps from: $${d\rho \over dz} = \sqrt{\left({\rho \over C}\right)^2 - 1}$$ to: $$\rho(z) = C \cosh\frac{z-z_0}{C}$$ I know that separation of variables is used so what I'm really asking is how to…
user1875290
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How to show $\int_{0}^{\infty}e^{-x}\ln^{2}x\:\mathrm{d}x=\gamma ^{2}+\frac{\pi ^{2}}{6}$?

How to show this equation below is true? $$\int_{0}^{\infty}e^{-x}\ln^{2}x\:\mathrm{d}x=\gamma ^{2}+\frac{\pi ^{2}}{6}$$ Where $\gamma$ is the Euler-Mascheroni constant....
esege
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Why this equation is true?

Pardon my ignorance. I don't know enough calculus to understand this. I assume this is a very easy question for this amazing site. I saw this on the The Theory of Riemann Zeta Function Book. $$\sum_{n=2}^{\infty}\pi…
esege
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Help in evaluating an Integral over an interval

So I have been given an Integral and its answer. $$\int_4^\infty\frac{1}{x^2+16}\,\text{d}x$$ The book used Trig substitution and got the answer: $${1\over 16}\left.\left(4\arctan\left({x\over 4}\right)\right)\right|_4^\infty$$*the last symbol…
Palwasha
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Proof of Integration formula

$$\int_0^{\infty}x^{-1}e^{-ax}\sin (bx) \;\mathrm dx = \arctan \frac{b}{a}$$ How to prove this result?
user71408
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Integrate the following function:

Evaluate: $$\int \frac{1}{ \cos^4x+ \sin^4x}dx$$ Tried making numerator $\sin^2x+\cos^2x$ making numerator $(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x$ Dividing throughout by $cos^4x$ Thank you in advance
chndn
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Determine the value of the integral $I=\int_{0}^{\infty} \frac{\ln\left(a^2+x^2\right)}{b^2+x^2}dx$

Determine the value of the integral $$I(a)=\int_{0}^{\infty} \frac{\ln\left(a^2+x^2\right)}{b^2+x^2}dx$$ My try: $\to I'(a)=\int_{0}^{\infty}\frac{2a}{(a^2+x^2)(b^2+x^2)}dx=\frac{\pi}{b(a+b)}$ Hence $I(a)=\frac{\pi}{b}\ln(a+b)+C$ Question: Find…
Iloveyou
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Prove that $\frac{d^n}{dx^n}\left(\frac{\sin x}{x}\right)=\frac{1}{x^{n+1}}\int_0^x y^n\cos\left(y+\frac{n\pi}{2}\right) \, dy,\: n\in \mathbb{N}$

Prove that $$\frac{d^n}{dx^n}\left(\frac{\sin x}{x}\right)=\frac{1}{x^{n+1}}\int_0^x y^n\cos\left(y+\frac{n\pi}{2}\right) \, dy,\: n\in \mathbb{N}$$ My try $n=0$ then $\frac{\sin x}{x}=\frac{1}{x} \int_0^x \cos y\,dy$ (true) Assuming $n=k$…
Iloveyou
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study the convergence of this integral $\int_{0}^{1}x^a|\ln{x}|^bdx$

let $a,b\in {\mathbb R}$, study the convergence of the following integrals: $$\int_{0}^{1}x^{a}\left\vert\,\ln\left(x\right)\,\right\vert^{b}\,{\rm d}x$$ My…
user94270
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Evaluating $\int_0^{\infty}\frac{\sqrt[3]{x+1}-\sqrt[3]x}{\sqrt x} \operatorname d\!x$

$$\int_0^{\infty}\frac{\sqrt[3]{x+1}-\sqrt[3]x}{\sqrt x}dx$$ I tried with $x=u^6$ then some trigonometric function and other thing but I faild what is your suggest to solve ?
user130806
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Volume of revolution over x-axis

Question: I need to find the volume of revolution of $$f(x)=\frac{2}{x+1},\;\; x\in [0,5],\;\;\text{about the x-axis}$$ In order to fully understand this question, one needs knowledge of understanding which shape to use to employ various methods…
Vanessa Vitiello
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Local integrability of two functions

Why $\log|x|$ is a locally integrable function and $1/|x|$ is not? Id know how their graphs look like but I don't know what is the exact difference causing local integrability of the first one.
luka5z
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