# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

2,855 questions
Filter by
Sorted by
Tagged with
10k views

• 1,746
2k views

• 53.5k
8k views

### Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx .$$ I tried to integration by parts and to get a pattern or to ...
• 7,110
1k views

38k views

### Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus

I need to find $\displaystyle\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}\ dx$ where $a > 0$. To do this, I set $f(z) = \displaystyle\frac{\cos z}{z^{2} + a^{2}}$ and integrate along the ...
• 6,558
3k views

• 261
7k views

### How to solve an definite integral of floor value function?

How do you prove this identity: $$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$ I'd very much appreciate your help on this one!
• 445
452 views

### Solving used Real Based Methods: $\int_0^x \frac{t^k}{\left(t^n + a\right)^m}\:dt$

In working on integrals for the past couple of months, I've come across different cases of the following integral: I\left(x,a,k,n,m\right) = \int_0^x \frac{t^k}{\left(t^n + a\right)^...
58k views

### Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
• 8,974
6k views

### Why it's true? $\arcsin(x) +\arccos(x) = \frac{\pi}{2}$

The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$ But I don't know how to explain it. I understand that the derivative of the equation is a ...
• 1,084
18k views

### Evaluate $\int_0^\pi xf(\sin x)dx$

Let $f(\sin x)$ be a given function of $\sin x$. How would I show that $\int_0^\pi xf(\sin x)dx=\frac{1}{2}\pi\int_0^\pi f(\sin x)dx$?
• 1,001
4k views

### Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24}$

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also didn'...
• 12.3k
2k views

### Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?

Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$ is independent on $a$?
• 1,563
12k views

### An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.

I want to prove: For an integrable function $f(x)$ and periodic with period $T$, for every $a \in \mathbb{R}$, $$\int_{0}^{T}f(x)\;dx=\int_{a}^{a+T}f(x)\;dx.$$ I tried to change the values and ...
• 7,110
2k views

• 583