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
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How to finish this integration?

I'm working with the integral below, but not sure how to finish it... $$\int \frac{3x^3}{\sqrt[3]{x^4+1}}\,dx = \int \frac{3x^3}{\sqrt[3]{A}}\cdot \frac{dA}{4x^3} = \frac{3}{4} \int \frac{dA}{\sqrt[3]{A}} = \frac{3}{4}\cdot\quad???$$ where $A=x^4+1$…
user984621
  • 333
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Substitution on integral

I have this (it looks simple) example $$\int \frac{3}{2-5x}\,dx$$ It looks really simple, no logarithms or trigonometric functions, but I just cannot get the proper result here...
user2932090
  • 323
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Reduction formula integral of $\cos^n(x)$

I need to find the reduction formula for the integral of $\cos^n(x)$. Ive split it into $\cos(x)\cos^{(n-1)}x$ in the hope of integrating by parts, but I'm unsure how to differentiate $\cos^{(n-1)}$, how should I proceed?
Jbarrell
  • 83
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Integration of $\sin(x)e^{-x^2}$

I recently came across how to $\int e^{-x^2}dx$ by "change of variable". i.e., let $I = \int e^{-x^2}dx$ then, $I^2 = \int e^{-x^2}dx\int e^{-x^2}dx$ and changing $x$ to $y$ in one of the integrals above to make it $I^2 = \int e^{-x^2}dx\int…
Yash
  • 281
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Reduction Formula without Partial Fractions

Let $n>1$ be an integer and $a \neq 0$ a constant. Prove the reduction formula $$\int \frac{1}{(x^2+a^2)^n} \, dx = \frac{1}{2a^2(n-1)} \left[ \frac{x}{(x^2+a^2)^{n-1}} + (2n-3) \int \frac{1}{(x^2+a^2)^{n-1}} \, dx \right].$$ Wanting to prove this…
Guest
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Could you help me with this integral: $\int\frac{\ln x-2}{x\sqrt{\ln x}} \, dx$?

$$\int\frac{\ln x-2}{x\sqrt{\ln x}} \, dx$$ Could I ask you, please, for helping me out with this example? I have some issues to get an result of this… Thank you in advance
user2932090
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Simple integration problem

I need to find the work done by some stationary particle on another. The work is defined as a scalar product of force and displacement ($W=\int_{}^{} \vec{F} \cdot d \vec{r}$). The force is inversely proportional to the square of distance between…
user147535
  • 23
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Calculating this integral: $I=\int_{0}^{\infty}(\log t)\,(\tan^2t)\,\mathrm{d}t$

How to calculate this integral? $$I=\int_{X_0}^{X}(\log t)\,(\tan^2t)\,\mathrm{d}t.$$ I tried integrate by parts and I found something related to: $$J=\int_{X_0}^{X}\dfrac{\log\cos t}{t^2}\,\mathrm{d}t.$$
Jika
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Integral of $\int\frac{dx}{x^2+1}$

I know the formula$$\int\frac{dx}{x^2+1} = \tan^{-1}(x) + C$$ But, when integrating by parts: $$u = x^2+1$$ $$u' = 2x$$ $$v' = 1$$ $$v = x$$ $$\int\frac{dx}{x^2+1} = uv - \int vu' dx = x(x^2+1) - \int 2x^2dx$$ $$ \int 2x^2dx = \frac{2x^3}{3} +…
stil
  • 383
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Differentiating an Integral of $f(x,t)$

How would I do the following calculation? $$ \frac{d}{dx}( \int_0^x{(x-t)f^{''}(t)dt}) $$ I tried it and I got $f^{'}(x)$, but I don't think I did it correctly.
pmal
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How to integrate $e^{t-u}du$

I'm not quite sure how to integrate $$\int e^{t-u}\,du$$ What particular rule do I use here? Thanks
Finance
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If $2xf ' (x) - f(x) = 0$ find $f$

So $2xf '(x) - f(x) = 0$ and we know that $f(1) =1$. So I actually need to find the integral of $2xf'(x) - f(x)$. Thanks.
Desepted
  • 23
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How to calculate an integral involving square roots and trigonometric functions?

I am waiting for the response since 2009. This question was asked by my teacher. $\displaystyle \int \mathrm{\sqrt{\sin{x}}}\,\mathrm{d}x$
Shruti Vaish
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Cases of Partial Fraction Decomposition

How many cases are there in integration using partial fractions?
anakin skywalker
  • 459
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Does it Make Sense to Use the Variable of Integration as a Bound?

I can't for the life of me seem to decide if using the variable of integration as a bound makes sense. For instance, integrating $y=x$ from $0$ to $x$. I don't think it does… But I'm not sure.
user82004
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