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
3
votes
5 answers

Simple integral by parts - can't solve it

How to solve this simple integral? $\int_a^b\cos\theta \sin\theta \,d\theta$ I tried integrating by parts: given the rule $\int u dv = uv - \int v du$ I substituted $\cos \theta$ as $dv $ and $\sin \theta$ as $u$ but I end up obtaining $0 = \sin…
John Smith
  • 735
3
votes
1 answer

How to take care of boundary term of integration by parts?

Let $f:[a,b]\to \mathbb{R}$ be a smooth positive function and let $\Omega = \{(x,y)\in\mathbb{R}^2\big|x\in[a,b],0\le y\le f(x)\}$. Consider a smooth function $F:\mathbb{R}^2\to\mathbb{R}^2$. Use integration by parts to show…
QD666
  • 557
  • 2
  • 9
3
votes
2 answers

Integrate the following $\int\frac{x+1}{x(1+xe^x)^2}dx$

Integrate the following $\int\frac{x+1}{x(1+xe^x)^2}dx$ I tried to multiply the numerator and denominator by $e^{-2x}$ Getting the following simplification $\int\frac{e^{-2x}(x+1)}{x(e^{-x}+x)^2}dx$ My next step is as follow…
Samar Imam Zaidi
  • 8,800
3
votes
2 answers

Double versus Iterated Integrals

I am having trouble discerning the difference between a double integral and an iterated integral. I have seen alternate notation to the effect of: \begin{align*} \iint\limits_R f(x,y) \ \ dR = \int_a^b \int_c^d f(x,y) \ \ dx \ dy = \int_a^b…
user465188
3
votes
2 answers

An integration that wolfram cannot help me.

$$\int e^{x\sin x+\cos x}\frac{x^4\cos^3 x-x\sin x+\cos x}{x^2\cos^2x}dx$$ I noted the fact that $\frac{d(x\cos x)}{dx}=-x\sin x+\cos x$ but I cannot apply the substitution on it.
yuanming luo
  • 651
  • 3
  • 15
3
votes
1 answer

Finding $f(2019)$ in definite integration

If $\displaystyle f(n)=\int^{1}_{0}(1+x+x^2+\cdots +x^{n-1})(1+3x+5x^2+\cdots +(2n-1)x^{n-1})dx$. Then $f(2019)$ is What I tried: $$1+x+x^2+\cdots +x^{n-1}=\frac{1-x^n}{1-x}$$ and $$1+3x+5x^2+\cdots…
jacky
  • 5,194
3
votes
0 answers

Closed form Integral of $\int \frac{dx}{x^2 + a x^b}$

Is there a closed form for the following integral (with $a$ and $b$ are given constants)? $$\int \frac{dx}{x^2 + a x^b}$$
Trung Phan
  • 81
  • 2
3
votes
5 answers

Integrate $\int \frac{\sqrt{x^2-1}}{x^4}dx$

I am trying to integrate $\int \frac{\sqrt{x^2-1}}{x^4}dx$ via trig substitution. I decided to substitute $x = \sec\theta$ into the square root and $dx = \sec\theta \tan\theta\,d\theta$. $$\int \frac{\sqrt{\sec^2 \theta-1^2}}{\sec^4\theta} \,dx =…
Evan Kim
  • 2,399
3
votes
0 answers

Finding values of A and B where the integral can be expressed in term of known functions

My problem is For what values of $A$ and $B$ can this integral be expressed in terms of known or elementary functions? $$\int \frac{1+(Ax)^{2B}}{x\sqrt{(x^{2B})-[1+(Ax)^{2B}]^{2}}} dx$$ I have tried integration by substitution of $u=1+(Ax)^{2B}$ and…
user134590
  • 63
3
votes
2 answers

Confusion with integrating sin(nx)sin(mx) and Kroenecker delta

The specific integral I'm working with is the following: $$ \int_0^a\sin(n\pi y/a)\sin(n'\pi y/a) $$ This is supposed to come out to $0$ in the case that $ n \neq n' $ and $\frac{1}{2}a$ in the case that $n= n'$. I can obtain this…
Bookie
  • 367
  • 3
  • 12
3
votes
2 answers

Basic Integration Problem: $\int \frac{di}{i-V/R}$

This is my first time using the maths stack forum, I normally use the electrical engineering stack. I am having difficulty taking the integral of a term (see the following link). $$\int \frac{di}{i}=\ln(i)+C \tag{1}$$ $$\int…
David777
  • 161
3
votes
3 answers

Is there any other solution for $f(x)=\int_0^\pi t f(t)\,\mathrm{d}t+\cos x $?

Is there any other solution for $f(x)=\int_0^\pi t f(t)\,\mathrm{d}t+\cos x $? I found one solution as follows but I have no clue how to prove that this solution is unique. Let $\int_0^\pi t f(t)=\lambda$, we have \begin{align} \lambda &=\int_0^\pi…
Display Name
  • 2,715
3
votes
2 answers

Show that $\forall s \in \mathbb R, \frac{2}{\pi}\int\limits_{0}^{+\infty} \frac{1 - \cos (st)}{t^2} dt = |s|$

Knowing that $$\int_{0}^{+\infty} \frac{1 - \cos (t)}{t^2} dt = \frac{\pi}{2}$$ Show that $$\forall s \in \mathbb R, \frac{2}{\pi}\int_{0}^{+\infty} \frac{1 - \cos (st)}{t^2} dt = |s|$$ Let $s \in \mathbb R$. If $s=0$, the formula is right. Else,…
anni
  • 919
3
votes
2 answers

Integrate using trigonometric substitution. Am I on the right path?

I have been trying to solve: $$\int \frac{\sqrt{x^2-9}}{x^3} dx$$ I am letting $ x = 3\sec \theta$ and so $dx = 3 \sec \theta \tan \theta$ So then I have: $$\int \frac{\sqrt{9\sec^2 \theta - 9}}{27 \sec^3 \theta} dx$$ $$\int \frac{\sqrt{9(\sec^2…
Jwan622
  • 5,704
3
votes
1 answer

Integral and measurability of a function

Consider the function $f:[0,1] \rightarrow [0,\infty)$ that is defined as follows: $$f(x) = 0 \text{ if $x$ is rational and } 2^n \text{ when $x$ is irrational}$$ Here $n$ is the number of leading zeros in the decimal expansion of $x$ and it can…
user62089
  • 1,356
1 2 3
…
99 100