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
1 answer

How to prove the following equation

This problem starts as a 1st order ODE where i need to solve for $\phi(t)$. $$\frac{d\phi(t)}{dt}-D\phi(t)=-f(t)$$ with a final condition $\phi(T)=1$. $D$ is a constant. Using an integrating factor $I(t)=e^{-Dt}$ to solve the ODE for $\phi(t)$, I…
Gnowl
  • 3
0
votes
2 answers

Evaluate the integral $\int_{0}^{\infty}e^{ax^3+bx^2}\,\mathrm{d}x$

Is there a closed form for $$\int_{0}^{\infty}e^{ax^3+bx^2}\,\mathrm{d}x $$?
user85798
  • 1
0
votes
1 answer

Change variables of triple integrals

I'm trying to solve this problem, the following is what I have done: Can anyone please tell me how to change it to (θ,φ,z)? Thanks!
user133140
  • 885
0
votes
2 answers

Double integral of $e^{x^2+y^2}dydx$?

I'd like to know the detailed solution of $$\int_{x=0}^{x=1} \int_{y=0}^{y=2} e^{x^2+y^2} dy dx.$$ I know I must find $\int_{y=0}^{y=2} e^{x^2+y^2}dy$, first but I don't even know how to start it.
Anonymous Guest
  • 11
0
votes
2 answers

What is the anti derivative of this definite integral?

What is the integral of the the absolute value of x over x dx, when the lower bound is -1 and the upper bound is 2?
Hannah
  • 67
0
votes
1 answer

How to evaluate this integral

Can some one help me evalute this integral or explain a bit where Pa(P) is binomial distribution with a formulae please help me evaluate this. the value for c = 2.
ALi
  • 1
0
votes
1 answer

Integrating problem with $\sin x$

I'm trying to integrate the following definite integral: $$\begin{align}\int_{0}^{\pi}{\sin x}\,dx\end{align}$$ My try: $$\begin{align}\int_{0}^{\pi}{\sin x}\,dx=\\\left(-\cos \pi\right)-\left(-\cos 0\right)=\\-1-(-1)=\\0\end{align}$$ But the actual…
Mohammad Fakhrey
  • 1,415
0
votes
1 answer

to prove that $F(c)=3c^2$

F(x) is a continuous function on [0,1], Such that $ \int_0^1 F(t)\,dt$=1. Prove that there exists a number c $\in$ (0,1) such that F(c)=3$c^2$.
Topology
  • 1,566
  • 11
  • 27
0
votes
1 answer

Definite integral question here?

In my book there is an example where the definite integral of $$\int\limits_0^{2\pi} \sin^2(nt) dt = \pi-\left[\frac{\sin(2n2\pi)}{4n} \right] $$ Then it says : $π$-$[sin(2*n*2π)/4*n ] $ = π Why? If n is even,then yes, $[sin(2*n*2π)/4*n ] $ = 0…
fdxsd
  • 39
0
votes
2 answers

Triple Integral with bounds in first octant

I am really confused on how to get my integrating function because I don't know, even after graphing, how the tetrahedron intersects the x-y-z axis. I am supposed to find the triple integral for the volume of the tetrahedron cut from the first…
Raynos
  • 1,623
0
votes
1 answer

Suppose $f$ is twice differentiable function such that ...

I am stuck with the following problem : I did integration by parts which gives the result $\,\,f'(1)$. Can someone explain? Thanks in advance for your time.
learner
  • 6,726
0
votes
1 answer

hyperbolic functions type problem

Here below is a problem which is difficult for me. Show that $$\int_{-\infty}^{\infty}(1-\frac{\tanh(x+y)}{\tanh(x-y)})dx=4y\coth{y}$$ Thank you!
Roger209
  • 1,233
0
votes
2 answers

Can this simple Integral be done analytically?

Anybody could help with some demonstration? $ \int_0^a \int_0^a e^{ik_1x_1} e^{ik_2x_2} e^{-q|x_1-x_2|} dx_1dx_2 $
lorniper
  • 225
0
votes
1 answer

Introduce a parameter to determine the value of $\int_0^1\frac{\log(1+x)}{1+x^2}dx$

How could I introduce a parameter to determine the value of $\int_0^1\frac{\log(1+x)}{1+x^2}dx$?
M.González
  • 214
0
votes
1 answer

Inequalitie of definite integral

I encountered the following integral question: Show that $$\frac{3}{8} \leq \int_{0}^{\frac{1}{2}} \frac{\sqrt{1-x}}{\sqrt{1+x}} \, dx \leq \frac{\sqrt{3}}{4}$$ I know I can just integrate it, but is there a way to obtain the result above without…
Kevin
  • 5
  • 1
1 2 3
…
76 77