Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

20559 questions
0
votes
2 answers

On the integral $\int_{0}^x\frac{\cos(z)}{e^{\frac{1}{z}}+1}dz$

The odd part of this peculiar integral $$f(x)=\int_{0}^x\frac{\cos(z)}{e^{\frac{1}{z}}+1}dz$$ seems to be equal to half the sine function, that is $$f(x)-f(-x)=\sin(x)$$ I have only observed this graphically through Desmos, but could anyone explain…
aleden
  • 4,007
0
votes
1 answer

Help with basic Double Integral

Given $$ I = \int_t^T(X(t,s)\int_t^sX(t,u)du)ds, \;\;\text{how do I get }\;I = \frac{(\int_t^TX(t,u)du)^2}{2}. $$ Many thanks for helping! And pardon me if this is straightforward.
Innocent
  • 609
0
votes
1 answer

Creating exponential decay function $f$ such that $\int_0^1 f = 1$

I am working on a project where I would like to take a weighted average of values in a set, with the weights decreasing exponentially. To do this, I am trying to find a function $f$ such that $\int_0^1 f = 1$. $f$ will be used in the following…
Erik
  • 103
  • 3
0
votes
2 answers

Property of definite integral

What is $$F'(x)\text{ if } F(x)=\int_{0}^{x}g(t)dt$$ I was thinking that $$F'(x)=g(t)\big|_{0}^{x}$$ is this correct?
Numbers
  • 539
0
votes
2 answers

Derivation of definite integral

Just a basic question to do with the derivation of the definite integral. When using the areas of n rectangles to approximate the area under a given curve, why does the sum of the areas of n+1 rectangles give a more accurate approximation? I can see…
stochasticmrfox
  • 909
0
votes
1 answer

Integrate $x^2/\ln(1+x^2) $ from $0$ to $1$?

Integrate: $$\int_0^1 \frac{x^2}{\ln(1+x^2)}\;dx$$ How do I calculate this?
MrJam
  • 1
0
votes
0 answers

Calculus: Work problem

A cylindrical-shaped tank is filled with water at the level of 1 foot. Its height is 8 feet, diameter is 6 feet, and weight density of water is 62.4 lb-ft^3. I've attempted to solve the work required to fill the tank to half, from the bottom. From…
RochNoure
  • 23
0
votes
1 answer

Finding value of $\int^{2}_{0}\frac{\ln(1+x)}{1+x^2}dx$

Finding value of $\displaystyle \int^{2}_{0}\frac{\ln(1+x)}{1+x^2}dx$ Try: Let $\displaystyle I(a) = \int^{2}_{0}\frac{\ln(1+ax)}{1+x^2}dx$ Then $\displaystyle I'(a) = \int^{2}_{0}\frac{1}{(1+ax)(1+x^2)}dx$ $\displaystyle =…
DXT
  • 11,241
0
votes
3 answers

Solve definite integral - solve analytically

I would need to solve this integral analytically using also a software. Can you help me? $$\int_{-2}^{x}(x-\varepsilon)\frac{(\varepsilon+2)^{1.5}(2-\varepsilon)^{1.5}}{2.5^{-2}} d\varepsilon - x = a$$
Marco
  • 147
  • 5
0
votes
0 answers

Integration question?

This question was asked in my exam. I couldn't solve it yet. I tried with substituting $1-x^3 = u$ but couldn't solve it.
Chirag Mahajan
  • 61
  • 6
0
votes
1 answer

Surface area of revolution of trig fn

I want to fine the surface area of revoultion around the y-axis of $$x=cos^3(\theta), y=sin^3(\theta)$$ I looked up a answer below but it's confusing me. (note that it's revolution around x-axis) To find the surface area, shouldn't it be $…
NK Yu
  • 263
  • 1
  • 8
0
votes
1 answer

A car moves in a straight line with velocity $t^{−2}−1/25 ft/s$.

For the total displacement, I keep getting $7/150$ but it seems to be incorrect! As for the total distance, I split the integrals to $[3,5]$ and $[5,6]$. I added the integrals together and took the absolute value and I still get $7/150$. Can anyone…
BeeliBala
  • 63
  • 6
0
votes
1 answer

Decreasing function with a target integral?

Problem Definition: I have a certain amount (S) of time that I can use over a fixed number of turns (T) in a game. I want to invest more of my time in the early turns, but still have a little time left for my last turn. As an example, if S = 55 and…
benjaminjsanders
  • 103
0
votes
0 answers

Square of an Integral into Single Integral

I have an integral of the form \begin{align*} \left(\int_{0}^{1}\mathrm{exp}(-t^2) \cos t^2 ~dt\right)^{2}. \end{align*} Is there any way to represent it into single integral form?. \begin{align*} \left(\int_{0}^{1}\mathrm{exp}(-t^2) \cos…
skorpion
  • 71
  • 7
0
votes
1 answer

Show $\int_0^1 (-\ln t)^{x-1} \, dt = \int_0^\infty t^{x-1}e^{-t}\,dt$

$$\Gamma(x)=\int_0^1 (-\ln(t))^{x-1} \, dt = \int_0^\infty t^{x-1}(e^{-t})\,dt$$ How do I show that these two integrals are equal to one another? Is there a way to use one of the integrals to derive the other?
Joshua Sin
  • 1
  • 1
1 2 3
…
76 77