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.

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How to integrate $\frac{x^2}{x^2+9}$?

I know you can use division to get $1-\frac{p}{x^2+9}$ and then use the result for arctan, but I was trying to do this using substitution and nothing seems to work? Is there a substitution that could be used for this?
Meep
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$\arctan(x) + \arctan(1/x)$ integration

How do I integrate $$\int_{1/b}^{b} \frac {\arctan(x)+ \arctan\left(\frac{1}{x}\right)}{x} dx \text{ ?} $$
Stuart Miller
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Expanding an Integrand

I am trying to solve this integral but it looks like I do not get the right result. Can you please help me? $$\int{(t-4)(t+2)^{\frac 4 5}}dt$$ I set $u=t+2$, so I get $\int{(u-6)u^{\frac 4 5}} \Bbb d t$ and then the solution I get is $$\frac 5 {14}…
ocram
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Variable in Feynman Integration

Evaluate: $$I=\int_{0}^{\frac{\pi}{2}} \ln(2468^{2} \cos^2x+990^2 \sin^2x) .dx$$ The suggested solution: $$f(y) = \int_{0}^{\pi/2} ln( y^{2}cos^{2}x + sin^{2}x)$$ Here- $ \frac{2468}{990} = y$ $$f'(y) = 2y \int_{0}^{\pi/2}\frac{cos^{2}x}{sin^{2}x +…
User1234
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How to calculate the integral of $f(x)$?

Let $f(x)$ be a function which satisfies the following two properties: 1) $f(x) + f(-x) =2$ 2) $f(1-x) = f(1+x)$ I need to calculate the $\int_0^{2016} f(x)dx$. I already tried to find $f(x)$ explicitly, but failed to do it. I don't know how to…
Rotem ben
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Calculate $\int_{0}^{3}{\left(\frac{12}{x^2 - 6x + 12}\right) \,dx}$

$$\int_{0}^{3}{\left(\frac{12}{x^2 - 6x + 12}\right) \,dx}$$ Assume that $x^2 - 6x + 12 = (x - 3)(x - 3) + 3 = (x - 3)^2 + 3$, then $t = x - 3 \rightarrow dt = dx$, since $$\int_{0}^{3}{\left(\frac{12}{x^2 - 6x + 12}\right) \,dx} = \int_{0}^{3}…
user155971
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How to compute $\int{\frac{5x^3+8x^2+x+2}{x^2(2x^2+1)}} dx$?

$$\int{\frac{5x^3+8x^2+x+2}{x^2(2x^2+1)}} dx$$ So ... how do I start? Numerator cant be factorized it seems, and this looks like a complicated expression ... I tried expanding the denominator to see if integration by substitution will work, but it…
Jiew Meng
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Integrate $\sqrt{x^2 - 2x}$

$$\int{\sqrt{x^2 - 2x}}$$ I think I should be doing trig substitution, but which? I completed the square giving $$\int{\sqrt{(x-1)^2 -1}}$$ But the closest I found is for $$\frac{1}{\sqrt{a^2 - (x+b)^2}}$$ So I must add a $-$, but how?
Jiew Meng
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Finding $\int_{0}^{\infty }\frac{1}{1+x^4}dx$

finding $$\int_{0}^{\infty }\frac{1}{1+x^4}dx$$ My attempt is: let $x=\sqrt{u}$ $dx=\frac{1}{2\sqrt{u}}$ $$\int_{0}^{\infty }\frac{1}{2\sqrt{u}(1+u^2)}du$$ here I stopped because I don't know how to complete this solution. Any help please.
E.H.E
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How can I prove that $ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c $?

How can I prove that $$ \int \text{sech}(x) ~ \mathrm{d}{x} = {\sin^{-1}}(\tanh(x)) + c? $$ I don’t know how to prove this identity. Any help? I tried to multiply by $ \dfrac{\cosh(x)}{\cosh(x)} $, and everything is okay, but at last I didn’t get…
E.H.E
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Using substitution to make an integral trivial

Consider the integral $$ \int_0^\pi \frac {\cos(\theta)} {f(\sin(\theta))}d\theta $$ Assume that $f(\sin(\theta))$ is nonzero on $[0,\pi]$. Can we use the substitution $u=\sin(\theta)$ to make the integral trivial ($\int_0^0$) and get this is zero…
Matt
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How to solve $\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}dx$?

I need to compute $$\int \frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}\ dx.$$ I tried it on wolfram but it timed out, maybe because I am on a mobile device. Any hint is appreciated.
Rohinb97
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Is it possible to calculate for example $\int_{0}^{1} x \mathrm{d}2x$

My question is just for fun, but I want also to verify if I understand something in variation calculus... I want to know if it is possible to calculate this : $$ \int_{0}^{1} x \mathrm{d}2x $$ A geometric argument is enough to conclude the area is…
ParaH2
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Trigonometric integrals

How do I evaluate this indefinite integral ? Integral $$\int\frac{x^2+n(n-1)}{(x\sin(x)+n\cos(x))^2}dx$$ What type of integral is it ? Is there any intuition involved in the approach to solve it? Edit: The complete term in denominator has 2 as…
user231156
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Given an integral equation, integrate the function.

$$f(x)=x+\int_0^1t(x+t)f(t){\rm d}t$$ Then what is $$\eta=\int_0^1f(x){\rm d}x$$ Ok you can write: $$\eta=\int_0^1\left(x+\int_0^1t(x+t)f(t)dt\right){\rm d}x=\frac12+\int_0^1\int_0^1t(x+t)f(t){\rm d}t{\rm d}x$$ How to eliminate f?
RE60K
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