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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Why does the initial u and v choices differ for u and v for cyclic integration by parts?

For context, the following is the cyclic integration by parts problem: $$ I = \int \sin (2x) \cos (3x) dx $$ After first setting $u$ as $\sin (2x)$ and $dv$ as $\cos (3x)$, I get the following expression: $$ I = \frac{1}{3} \sin(3x)\sin(2x) -…
JeyuLeoChou
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I don't understand why the derivative of $f(x) = \int_2^{-x+3} e^{t^2} dt$ is negative.

Someone asked me about the following exercise and I tried my best to help them but couldn't understand this. Given: $$f(x) = \int_2^{-x+3} e^{t^2} dt$$ According to wolframalpha, the derivative is: $$f'(x) = -e^{(3-x)^2}$$ But I cannot for the life…
Ian Rehwinkel
  • 155
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How can I prove exercise 89, section 5.5 from Stewart, Calculus - Early transcendentals (7th ed)

For $a$ and $b$ positive, prove that: $$\int_0^1 x^a(1-x)^b dx = \int_0^1 x^b(1-x)^a dx $$ I've tried: $$ u = 1-x, du = -dx $$ $$ x = 1 - u $$ $$ \int_1^0 (1-u)^au^b (-du) = \int_0^1 u^b (1-u)^a du $$ Now I can't think. What should I do? I have no…
Gabriel Bento
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Concept Of Double Integration

Can someone explain how double integration is equivalent to calculating volume as single integration is calculating area.
Nithin Jose
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Help Visualizing the double integral

I understand that the double integral is However what confuses me is when I try to visualize why this formula only accounts for the region inside the bounds and not the whole rectangular region. My guess is it has to do with one of the bounds…
CatsOnAir
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Solving $\int_{x_{1}}^{x_{2}}\frac{dx}{\sqrt{E-U_{0}\tan(ax)^{2}}}$

Please help me to find this integral without long calculations. My attempt took several several sheets, and the answer is frighteningly piled up. Although I have come to success, I would like to see a more accessible…
George W
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Solve $\int \frac{\sin(\pi x) }{\ln x}dx$

Wolfram Alpha thinks that it is unsolvable, but... My solution so far: $$\int \frac{\sin(\pi x) }{\ln x}dx = \int \frac{e^{i\pi x} -e^{-i\pi x}}{2i\ln x}dx.$$ Let us split sum and focus on the first integral, using the Feynman's technique: $$\int…
user184868
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How to derive the integral $ I = \int^{\infty}_{0} 1-\left(\frac{x^a}{c + x^a}\right)^ndx, n\in \mathbb{Z}^+$

$a,c \in \mathbb{R}$ and $a > 1, c > 0$. I already know the answer includes $\Gamma(n+1/a)\Gamma(1-1/a)/\Gamma(n)$. But I couldn't get how integral like this can be expressed by gamma function since gamma function includes exponential function…
LOREY CHU
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Integration Question Dependence on Unknown Values

The question given is the following: Let $\Omega$ be the domain bounded by the paraboloid z = 4 - ($x^2$ + $y^2$) and the plane $z = 0$. Let $f$ be the scalar field $f(x,y,z) = ax + by + cz$, where a, b and c are constants. Find $∫_{\Omega}$ $f$…
Sam Connell
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Unable to find my mistake in the process of solving a seemingly simple integration problem

I am solving the following integration problem. I got two results when I followed two different methods. But I know one is wrong, but I am not sure where my mistake is. I request someone to help me find the mistake. $$\int \frac{{\mathrm…
SKGadi
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How do I calculate a sum $\sqrt{\frac {\hbar{(v)} }{4G}} r$ with respect to a changing $v$?

How do I calculate a sum based on the variable $v$ in this expression $\sqrt{\frac{\hbar(v) }{4G}} r$ which changes in increments of 1 with $v_i= 1$ to $v_f = 6.25E34$? Straight forward it would look something like this: $$\sqrt{\frac…
Tivity
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Integration that I coudn't figure out

I am not mathematician but an engineer. At some stage of derivation for a formulation I've just been stuck with the following integration which might be super easy and stupid question for all of you as a professional mathematician but your help is…
Cagdas Akalin
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Two definitions of line integral

I've seen in some textbooks, the line integral is defined as $\displaystyle \int_\gamma Pdx+Qdy$, where $\gamma$ is a path and P and Q are continuous functions.However, in other books the line integral is defined as $\displaystyle\int_a^b…
user533661
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Does this definite integral exist?

So I have the following definite integral: ${\int_{0}^{1}x(2x^2-1)^{-10}}dx$ I suppose I cannot just integrate over the interval (0,1) because of the discontinuity there. I used the substition method where $t=2x^2-1$ and then the new integral…
kuchejdatomas
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