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.

73636 questions
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The antiderivative of $\cos^5(x)\sin^5(x)$ - is this incorrect?

I like to check my answers with wolframalpha, and this one's stubbornly coming up as false when set equal to its answer for the antiderivative, but I can't figure out where I'm going wrong. Using the identity $\cos^2x = (1-\sin^2x)$, I rewrote the…
Calculistening helplessly
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Integral calculation

I have a power spectral density function of the form $$S(\omega) = \frac{a}{(\omega^2-K^2)^2 + K^2 h^2 \omega^2},$$ in which $a, K, h$ are some positive constants. I want to calculate the corresponding spatial correlation function which is defined…
Shahram Khazaie
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Show that $\int_{0}^{1}{\frac{\sin{x}}{x}\mathrm dx}$ converges

As title says, I need to show that the following integral converges, and I can honestly say I don't really have an idea of where to start. I tried evaluating it using integration by parts, but that only left me with an $I = I$ situation. $$\int…
user93603
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How prove this $I=\int_{0}^{\infty}\frac{1}{x}\ln{\left(\frac{1+x}{1-x}\right)^2}dx=\pi^2$

Prove this $$I=\int_{0}^{\infty}\dfrac{1}{x}\ln{\left(\dfrac{1+x}{1-x}\right)^2}dx=\pi^2$$ My try: let $$I=\int_{0}^{\infty}\dfrac{2\ln{(1+x)}}{x}-\dfrac{2\ln{|(1-x)|}}{x}dx$$
math110
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What is the integral of $e^{-x^2/2}$ over $\mathbb{R}$

What is the integral of $$\int_{-\infty}^{\infty}e^{-x^2/2}dx\,?$$ My working is here: = $-e^(-1/2x^2)/x$ from negative infinity to infinity. What is the value of this? Not sure how to carry on from here. Thank you.
lakshmen
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Slope function of a curve

I have a question I am having trouble answering: The slope function of a curve is: $$\frac{dy}{dx}=ux+k$$ u and k are constants. The curve passes through $(0,-1)$ and $(2,-5)$ At $(2,-5)$ the slope equals $1$. How do I get the equation? I have tried…
user
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Mellin transform definition

is thre a possible meaning or definition to the Mellin transform $$ \int_{0}^{\infty} \frac{dt}{t-1}t^{s-1}= F(s) $$ i know that $$ \int_{0}^{\infty} \frac{dt}{t+1}t^{s-1}= F(s)= \frac{\pi}{sin(\pi s)} $$ howver can one overcome the pole at $ t=1 $…
Jose Garcia
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Assuming two functions are the same

If we know the following: $$\int_a^b f(x) \,dx= k$$ and: $$\int_a^b g(x) \,dx= k$$ Can we deduce that $f(x) = g(x)$? If not, could someone find me an example where the above is the case and $f(x)$ does not equal $g(x)$? Thanks in advance.
Nav Bhatthal
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Is the following statement true? $\int_{-1}^{x}\ {2x+x^2}\,dx = \int_{-1}^{x}\ F'(x) = F(x) = \int_{-1}^{x}\ {2t+t^2}\,dt$

I have trouble understanding easy integral concept. Here, one of the integral theorem states: And I was wondering what would happen if $F'(x)$ is back in to the integral. For example, $F(x) = \int_{-1}^{x}\ {2t+t^2}, x \in [-1,5]$ $$F'(x) = 2x +…
Belphegor
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Evaluating $\int^\infty_0 \frac{\tanh(x)}{x\cosh(2x)}dx$

I am trying to evaluate $\int^\infty_0 \frac{\tanh(x)}{x\cosh(2x)}dx$ by Feynman's technique. Let $I(b)=\int^\infty_0\frac{\tanh(bx)}{x\cosh(2x)}dx$, then $I'(b)=\int^\infty_0 sech^2(bx)sech(2x)dx$. However, I am unable to find out the integral in…
HeyFan
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Evaluating the integral: $\int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx$

I am interested in evaluating the following integral: $$ \int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx $$ Using Matlab, Numerically it seems that the integral is convergent, but I'm not sure about it. How can we prove that the integral…
Shahram Khazaie
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Find the following integral: $\int {{{(\ln x)}^2}} dx$ by using the method of integration by parts

Find $\int {{{(\ln x)}^2}} dx$ by using the method of integration by parts. My attempt: $$\eqalign{ & \int {{{(\ln x)}^2}} dx = \int {2\ln x} dx \cr & u = \ln x,{\rm{ }}{{du} \over {dx}} = {1 \over x} \cr & {{dv} \over {dx}} = 2,{\rm{…
seeker
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$ \int_0^1 |f(x)-t| \, dx \le \frac{(1-t)^2+1}{2}$

Let $ f(x)>0$, $f''(x)>0$, and $ \int_0^1 f(x)\,dx=1 $, for $t\in \mathbb R $, prove that: $$ \int_0^1 |f(x)-t| \, dx \le \frac{(1-t)^2+1}{2}.$$ This inequality maybe is very interesting. But I can't prove this. sorry,everyone ,This Problem…
math110
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Using the definition of the integral, find $\int_0^a x^2 dx$

The definition of the integral I was given (which after searching around seems like the common definition) is the value of the inf{upper sums across all dissections} (integral exists when this coincides with the sup{lower sums across all…
user68293
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Can an indefinite integral have multiple answers? (Besides the ' + C')

So I came across with this integral today in my midterm: $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2 $$ And I got two correct answers: $$\frac {\sec^2(\pi x)}{4\pi} +C$$ And $$\frac {\tan^2(\pi x)}{4\pi} + C$$ The first one, I get it by substituting…
ChairOTP
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