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
1
vote
1 answer

Seeking to evaluate $\int_{0}^{-1}\frac{e^{ax}+\frac{1}{a}xe^{a/x}-1}{x}\mathrm dx$

We wish to evaluate this integral, $$I=\int_{0}^{-1}\frac{e^{ax}+\frac{1}{a}xe^{a/x}-1}{x}\mathrm dx, a\ge1$$ We try: $$I=\int_{0}^{-1}\left(\frac{e^{ax}}{x}+\frac{1}{a}e^{a/x}-\frac{1}{x}\right)\mathrm…
user569129
1
vote
2 answers

Change of variable in integrals

I am trying to solve a definite integral of a positive function, but I keep getting 0.
mike
  • 667
1
vote
1 answer

Proof of volume of sphere using Definite Integration

Prove that the volume of the sphere is $\frac{4}{3}\pi R^3$ We can think of a sphere as made up of infinite number of concentric shells of thickness $dr$ and volume $4\pi r^2dr$ So the total volume of sphere is equal to $$\int_0^R4\pi r^2dr =…
user677491
  • 333
1
vote
2 answers

A definite integral: $\int_{0}^{\pi/2} \frac{\sin x~\mathrm dx}{\sin x+\cos x+ e^x}$

Mathematica can do this integral, $$\int_{0}^{\pi/2} \frac{\sin x~ \mathrm dx}{\sin x+\cos x+ e^x}\,,$$ the question is: how to do it by hand?
Z Ahmed
  • 43,235
1
vote
2 answers

Identity regarding exponential integral

I came across an identity which is described in this book as "classical" (page 85/350): \begin{equation} \int_{0}^{x} \frac{1 - e^{-y}}{y} dy = E_1(x) + \log(x) + \gamma, \qquad x > 0 \end{equation} where $\gamma$ is the Euler-Mascheroni constant…
tosi3k
  • 654
1
vote
3 answers

If $f(x)=\sum^{\infty}_{n=0}\frac{\sin(nx)}{4^n}$. Then value of $\int^{\pi}_{0}f(x)dx$ is

If $\displaystyle f(x)=\sum^{\infty}_{n=0}\frac{\sin(nx)}{4^n}$. Then value of $\displaystyle \int^{\pi}_{0}f(x)dx$ is Plan $$\int^{\pi}_{0}\sum^{\infty}_{n=0}\frac{\sin(nx)}{4^n}dx=\sum^{\infty}_{n=0}\int^{\pi}_{0}\frac{\sin(nx)}{4^n}dx$$ How do…
jacky
  • 5,194
1
vote
2 answers

Non Uniform Density Rod Center of Mass

Consider a rod in three dimensional space where $y$ is the height axis. $h$ is the height of the rod and $a$ is the radius of the rod. The function $\rho(r, \theta, y)$ is the density function. The mass of the rod can be calculated with $$ m…
fullnitrous
  • 19
1
vote
1 answer

Prove that $\frac{3}{4}<\int_{0}^{1} x^x dx <\frac{85}{108}.$

Let $I=\int_{0}^{1} x^x dx$, the function $f(x)=x^x$ attains a local minimum at $x=e^{-1}$ and $f''(x)>0, \forall x\in (0, \infty).$ So $f(x)>e^{-e^{-1}}\approx 0.69.$ Next, one may choose a point $x=1/2$, the chords in $x \in (0,1/2)$ and in $x \in…
Z Ahmed
  • 43,235
1
vote
2 answers

Prove $\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx$.

Let $f(x) \in C[0,1]$, and $f(x)>0$ over $[0,1]$. Prove $$\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx.$$ If we denote $$F(x):=\ln \int_0^x f(t){\rm d}t-\int_0^x \ln f(t){\rm d}t, ~~~x \in[0,1]$$ Differentiate the both sides with respect to $x$，…
mengdie1982
  • 13,840
  • 1
  • 14
  • 39
1
vote
1 answer

A family of four definite integral, are they doable?

Let $$I_{\mp n}=\int_{-1}^{1} \frac{x^2}{1\mp e^{\sin(x^n)}} \,dx,~~ n=1,3.$$ We can visualize four integrals: $$I_{+1}, ~~I_{-1}, ~~I_{+3}, ~~I_{-3}.$$ Are these four integrals doable and convergent?
Z Ahmed
  • 43,235
1
vote
1 answer

If $\int_0^t \cos(f(t)) \,dt =\frac{l}{v}$ and $\int_0^t \sin(f(t))\, dt =\frac{ut}{v}$, find $t$

While solving a physics problem I ended up with the following integration and I don't know how to solve it. Please help. $$\int_0^t \cos(f(t)) \,dt =\frac{l}{v} $$ And $$\int_0^t \sin(f(t)) \,dt =\frac{ut}{v} $$ We don't know what $f(t)$ is. We…
Abhinav
  • 450
1
vote
0 answers

If $\int_{0}^{\pi/2}f(\sin 2x) \, dx = k \int_{0}^{\pi/4} f(\cos x) \cos x \,dx$, find $k$.

Find $k$ such that $$\int_{0}^{\pi/2}f(\sin 2x) \,dx = k \int_{0}^{\pi/4}f(\cos x) \cos x \,dx.$$ I applied the properties of DI , but am not able to change $\sin 2x$ into $\cos x$.
maveric
  • 2,168
1
vote
1 answer

Fundamental Theorem of Calculus with two different letters

I got a little bit confused on this integral, and the reason is because I got 2 different letters on the function, it tells me to differentiate. $$F(x)=\int_{5x}^{6x+1}f(x+u)du$$ And I don't know what to do
Max Villegas
  • 99
1
vote
3 answers

Evaluate the following integral of $\int_1^e\sin{(\ln{(x)})}\mathrm{d}x$ without complex analysis

This appears in my guide of definite integrals ; I haven't seen complex analysis yet.. $$\int_1^e\sin{(\ln{(x)})}\mathrm{d}x\tag1$$ I used integration by parts and substitution 3 times in (1), but I get nowhere.. any help?
Sebastian Fernandez
  • 123
1
vote
0 answers

Finding force due to fluid pressure gives different answers

So there were two methods presented: the first one was using the $F = PhA$ where $P$ is density, $h$ is the height, $A$ is the area of the strip. The second one was using the center of mass which is $F = PHA$, where $H$ is the center of mass. I…
Bido262
  • 85
1 2 3
…
76 77