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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Integral of a product

Given the following integral: $$I(N)=\int_{0}^{t}\prod_{k=1}^{N}\sin(k\omega \tau)d\tau$$ does someone know if is it possible to find the solution of $I(N)$ in a closed form? I'm able to find the formula for $N=2$ $$I(2)=\frac{1}{2}\frac{\sin(\omega…
Riccardo.Alestra
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How to integrate this fraction: $\int\frac{1}{1-2x^2}dx$?

I'm not sure how to integrate this: $$\int\frac{1}{1-2x^2}dx$$ I think it has to be this: $$ -2\cdot \arctan(x)$$ Or this: $$\arctan(\sqrt{-2x^2})$$
user2952272
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Evaluation of the following integral

How do I evaluate the following integral? $$I=\int_{0}^{\infty}\frac{\sinh(a)k\ dk}{\cosh(k) + \cosh(a)}, \qquad a \geq0$$
chetan joshi
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Why no constant $C$ when integrating $\int_{0}^{x}f(t)dt$?

Why is there no constant $C$ added when integrating from $0$ to $x$ in $$ \int_{0}^{x}f(t)dt$$ Is this the only case when $C$ can be omitted?
mavavilj
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Integral of step functions. Doubts with Apostol's excercise

In the "Calculus" Apostol's Book in the part of the Integral of a Step Function The first exercise: $\int_{-1}^{3}[x]dx$ where $[x]$ is the integer part of $x$ the result is 2. (page 70). I could not get this result. I'm doing something wrong or…
nomol
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Finding primitives for Lebesgue integrable functions

I was wondering if there is a set of algebraic "rules" for finding primitives of Lebesgue integrals as there is one for finding primitives of Riemann integrals. I.e. for $x^{n}$ the primitive is $\frac{x^{n+1}}{n+1} + C_{0}$ in the Riemann world…
Frank
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Why can't I solve $\int (1+x^2)^{-1}dx$ this way?

Why can't I solve the integral $\int (1+x^2)^{-1}dx$ this way? Or at least, I've been told that this way is wrong, why is that? $$\int (1+x^2)^{-1}dx=\frac i 2\int (x+i)^{-1}dx-\frac i 2 \int (x-i)^{-1}dx=\frac i 2 \left(\ln\left(\frac…
YoTengoUnLCD
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Proof for formula $\int e^{g(x)}[f'(x) + g'(x)f(x)] dx = f(x) e^{g(x)}+C$

I recently saw someone using this formula here on one of the questions since I can't comment , can someone please give me the proof of this equation and type of problems where it can be used ?
user270337
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Calculate $\int {\frac{{\sqrt {x + 1} - \sqrt {x - 1} }}{{\sqrt {x + 1} + \sqrt {x - 1} }}} dx $

Calculate $$\int {\frac{{\sqrt {x + 1} - \sqrt {x - 1} }}{{\sqrt {x + 1} + \sqrt {x - 1} }}} dx $$ My try: $$\int {\frac{{\sqrt {x + 1} - \sqrt {x - 1} }}{{\sqrt {x + 1} + \sqrt {x - 1} }}} dx = \left| {x + 1 = {u^2}} \right| = 2\int {\frac{{(u…
Evgeny Semyonov
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How to find this indefinite integral?

Find $\displaystyle\int \frac{dx}{2\sqrt x+\sqrt{x+1}+1}$ I think I should change dx and let x = something. What is your suggestion?
joefu
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shortcut for sum of integral for computational optimisation?

$$\int_{x=0}^\tau \frac{1}{\|[\cos x,\sin x]-\vec{p}\|^n}$$ where $n$ is either $1$ or $2$ and $\vec{p}$ is a part of $\Bbb{R}^2$. Is there a way to simplify this into something that is much quicker for a computer to calculate or is the best option…
god of llamas
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How to approach an integral over $g(\cos(t))$ from $0$ to $2\pi$, where $g(x)$ is nasty?

For notational convenience, let $f(t) = a^2 + 2 a b \cdot \cos(t) + b^2$, where $a,b$ are both positive real constants and $t$ will be the variable of integration, which is supposed to be carried out from $t=0$ to $t=2 \pi$. I want to find an…
TriSSSe
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Maclaurin Series

Integrate using first three terms of appropriate series... $$\int_0^1 \sin x ~dx.$$ So I use $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!}$$ for my three terms and if I integrate just that I get the answer which is $.3103$. However the solution…
Chef Flambe
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Functions with logarithmic integrals

I'm self teaching from Core Maths for Advanced Level by Bostock and Chandler. They say this: $$\int \frac{1}{ax + b} dx = \frac{1}{a} ln |ax + b| + K = \frac{1}{a} ln A|ax + b|$$ There's no explanation of where the $A$ comes from, or where the…
PeteUK
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solve for $\int_{0}^{{\alpha}{b}}(a^x-1)dx=\int_{{\alpha}{b}}^{b}(a^x-1)dx$

I am sitting with a problem and my calculus is a bit (ok very) rusty. $\int_{0}^{{\alpha}{b}}(a^x-1)dx=\int_{{\alpha}{b}}^{b}(a^x-1)dx\\ 0<\alpha<1\\ b\geq1$ Solve for a. any help would be greatly appreciated...
Quantifeye
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