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
2
votes
2 answers

Reduction formulae question.

$I_n=\int_0^\frac{1}{2}(1-2x)^ne^xdx$ Prove that for $n\ge1$ $$I_n=2nI_{n-1}-1$$ I end up (by integrating by parts) with: $I_n =e^x(1-2x)^n+2nI_{n-1}$ I am not sure how $e^x(1-2x)^n$ becomes $-1$?
maxmitch
  • 651
2
votes
0 answers

$ \int_{0}^{\pi}\frac{x^2}{(1+x \sin x)^2}\,\mathrm dx$

$$\int_{0}^{\pi}\frac{x^2}{(1+x \sin x)^2}\,\mathrm dx$$ I got $$2I=\int_{0}^{\pi}\frac{x^2}{(1+x \sin x)^2}\,\mathrm dx + \int_{0}^{\pi}\frac{x^2}{(1 - x\cos x + \sin x)^2}\,\mathrm dx $$ After this step I got stuck... According to me I tried $\cos…
Diwakar yadav
  • 21
2
votes
2 answers

How would you integrate $\sin(1/x)$?

I have found out that if i use u-substitution here's what i get : $$\int ( \sin(u) ) \, dx$$ \begin{align*} u &= 1/x = x^{-1}, \\ du / dx &= -x^{-2}, \\ dx / du &= -x^2, \end{align*} $$\int ( -\sin (u)x^2) \, du$$ $$= \text{?}$$ as you can see…
sweet lovely
  • 25
2
votes
1 answer

Find the integral: $\int{\frac{6x^4-5x^3+4x^2}{2x^2-x+1}}dx$

I'm reading Differential and Integral Calculus by Piskunov, and have read up to integration using substitution. I've learned how to solve integrals of the form: $$\int{\dfrac{Ax+B}{ax^2+bx+c}}dx$$ The integrals of the above form are solved by…
csmathhc
  • 330
2
votes
1 answer

Calculate double integral with specific region

I need to calculate $$\iint\frac{dx\,dy}{\sqrt{x^2 + y^2}}$$ $$x^2 + y^2 < 2y$$ I tried to solve this via polar coordinates $$0 \leq r < 2\sin(\phi)$$ $$0 \leq \phi \leq 2\pi$$ So, our integral becomes $$\int_0^{2\pi} d\phi \int_0^{2\sin(\phi)} \,…
zigzig
  • 23
2
votes
1 answer

Triple integral between $z^2 = {(x - 1)}^2 + y^2$ and the unit sphere.

I have to compute $\int_D f$, where $D$ is the region in ${(0 , \infty)}^3$ between the cone $z^2 = {(x - 1)}^2 + y^2$ and the sphere $x^2 + y^2 + z^2 = 1$, and $f : D \to \mathbb{R}$ is given by $f(x , y , z) = z \sqrt{x^2 + y^2}$. My attempt: If I…
joseabp91
  • 2,360
2
votes
4 answers

Let $\int_{1}^{2} e^{x^2} \mathrm{d}x=a$. Then find the value of $\int_{e} ^{e^4} \sqrt{\ln x} \mathrm{d}x$

Let $\int_{1}^{2} e^{x^2} \mathrm{d}x =a$. Then the value of $\int_{e} ^{e^4} \sqrt{\ln x} \mathrm{d}x $ is (A) $e^4-a$ (B) $2e^4 - a$ (C)$e^4 - e - 4a$ (D) $2e^4-e-a$ $\bf{Try:}$ Let $\ln x =t^2$ then $x=e^{t^2}$ and $dx=2te^{t^2} dt$. Then the…
Math-Learner
  • 732
2
votes
3 answers

Solve $\int e^{x^2+x}(4x^3+4x^2+5x+1)dx$

One of the solutions tried to write the integral a $$\int e^{x^2+x}((2x+1)p(x) + p’(x))dx$$ Where $p(x)=2x^2+bx+c$ I have no idea why they chose the lead coefficient to be $2$. Going with this, $b=1$ and $c=0$ So $$\int e^{x^2+x}…
Aditya
  • 6,191
2
votes
3 answers

Integrating secant squared times tangent

Integrate the function. $$ \int \sec^2 x \tan x dx $$ I'm trying to get a proper substitution, but I couldn't get anything proper.
user76849
2
votes
2 answers

find this integral $\int_{0}^{+\infty}e^{-\beta {x}}\left(\frac{1}{x}-\coth{x}\right)dx$

find the intgeral $$\int_{0}^{+\infty}e^{-\beta {x}}\left(\dfrac{1}{x}-\coth{x}\right)dx$$ where $\Re{(\beta)}>0$ maybe useStarting with the infinte series expansion for $\arctan x$ $$\arctan x = \sum _{k=1}^{\infty } \frac{(-1)^{k-1} }{2 k-1}x^{2…
math110
  • 93,304
2
votes
1 answer

Integrating: cos(t)*(exp(cos(t)) + exp(sin(t))*cos(t))...

I'm trying to…
Leon
  • 1,117
2
votes
4 answers

How to calculate $\frac{d}{dx}\left(\int_{\sin(x)}^{\cos(x)}\:e^{t^2} \, dt\right)$?

How to calculate $\frac{d}{dx}\left(\int_{\sin(x)}^{\cos(x)} e^{t^2} \, dt \right)$? First of all, how do you even visualise this? What does it mean when you integrate from a function to another function? Do I apply the fundamental theorem of…
CountDOOKU
  • 1,065
2
votes
1 answer

Find $\underbrace{\int_0^1 \cdots \int_0^1}_{a+b \text{ times}} \frac{dx_1 \,dx_2 \cdots dx_{a+b}}{(1+x_1 x_2 \cdots x_a)(1+x_1 x_2 \cdots x_{a+b})}$

Can we find a closed form or reduce or even evaluate the below weird integrals? $$\underbrace{\int_0^1 \cdots \int_0^1}_{a+b \text{ times}} \dfrac{dx_1 \, dx_2 \cdots dx_{a+b}}{(1+x_1 x_2 \cdots x_a)(1+x_1 x_2 \cdots x_{a+b})}$$ I have tried to…
BooleanCoder
  • 862
2
votes
0 answers

Integrability of the function $f:I \rightarrow \mathbb{R}$.

I was attempting the following problem, I know there is an answer for this, I just want to make sure my effort is ok. For the generalized rectangle $I= [0,1]\times [0,1]$ in the plane $\Bbb R^2$ $$f(x,y)=\begin{cases} 5 & if\ \ (x,y)\ is\ in\ I\…
LakeHermanStoutMonster
  • 41
2
votes
1 answer

$ \int \frac{x-4}{\sqrt{x^2-4x+5}}\, dx$

I'm trying to solve this irrational integral $$ \int \frac{x-4}{\sqrt{x^2-4x+5}}\, dx$$ doing the substitution $$ x= \frac{5-t^2}{2 \cdot (2+t)}$$ according to the rule. So the integral becomes: $$ \int \frac{1}{2} \cdot…
Anne
  • 2,931
1 2 3
…
99
100