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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Evaluation of Integral $\int_{0}^1 \frac{\arctan x }{1+x} dx$

How do you compute $$\int_{0}^1 \frac{\arctan x }{1+x} dx$$
imranfat
  • 10,029
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Integral with constant u-substitution

This is a simple integral. $$ \int \frac{1}{3x}dx $$ with an equally simple solution of $$ \frac{1}{3}\ln|x| +c $$ My question is that if you chose to use u-substitution and used u = 3x, the solution appears to work out as follow: $$ \int…
Deeves
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Show $\lim_{n\to \infty}n\int_{0}^{\frac{\pi}{2}}(1-\sqrt[n]{\sin(x)})\,\mathrm{d}x = \frac{\pi \ln(2)}{2}$

Here is an interesting limit of an integral I do not know how to begin. Any help is greatly appreciated. $\lim_{n\to \infty}n\int_{0}^{\frac{\pi}{2}}(1-\sqrt[n]{\sin(x)})\,\mathrm{d}x$ I know it converges to $\frac{\pi \ln(2)}{2}$, but how?. Thanks…
Cody
  • 1,551
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Problem with calculation this integral: $\int_0^\pi \frac{dx}{1+3\sin^2x}$

Question Calculate this integral: $$\displaystyle\int_0^\pi \frac{dx}{1+3\sin^2x}$$ Solution $$I=\displaystyle\int \frac{dx}{1+3\sin^2x}=\displaystyle\int \frac{dx}{\cos^2x+4\sin^2x}=\displaystyle\int \frac{\sec^2x\;dx}{1+4\tan^2x}$$ Let's apply…
Micheal Brain Hurts
  • 2,380
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Computing $\int_0^\infty\frac{1}{(1+x^{2015})(1+x^2)}$

How can I compute $$\int_0^\infty\frac{1}{(1+x^{2015})(1+x^2)}\quad?$$ My attempt: Looking at the limits of the integration I see that we should induce some $\tan^{-1}(x)$ so if we put $\infty$ we would get something like $\frac{\pi}{2}$ . But I…
Archis Welankar
  • 15,910
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Evaluating $\int_{0}^{\infty}(\ln \tan^2 bx)/(a^2+x^2)\ dx$

Some time ago I came across one of the integrals, which still goes over my mind: $$\int_{0}^{\infty}\frac{\ln \tan^2(bx)}{a^2+x^2}dx$$ a and b are parameters. I would be interested in possible solutions with complex analysis and without it as…
Martin Gales
  • 6,878
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Integrate $\frac{5x^3 +2}{\sqrt{x^3+1}}$

$$\int\frac{5x^3+2}{\sqrt{x^3+1}}\,\mathrm{d}x$$ Sad to say this has really stumped me and nothing I have tried has worked. I used Wolfram Alpha to find that the answer is simply $2x\sqrt{x^3+1}$ but it says the method is unavailable and I have no…
user85798
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Gaussian Integral

Consider the following Gaussian Integral $$I = \int_{-\infty}^{\infty} e^{-x^2} \ dx$$ The usual trick to calculate this is to consider $$I^2 = \left(\int_{-\infty}^{\infty} e^{-x^2} \ dx \right) \left(\int_{-\infty}^{\infty} e^{-y^{2}} \ dy…
yahooguy
  • 109
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Integral of $\sin x \cdot \cos x$

I've found 3 different solutions of this integral. Where did I make mistakes? In case there is no errors, could you explain why the results are different? $ \int \sin x \cos x dx $ 1) via subsitution $ u = \sin x $ $ u = \sin x; du = \cos x dx…
Jimch
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Help in finding $\int \frac{x+x\sin x+e^x \cos x}{e^x+x\cos x-e^{x} \sin x} dx$

I want to find $$\int \frac{x+x\sin x+e^x \cos x}{e^x+x\cos x-e^{x} \sin x} dx.$$ But since algebraic, exponential and trigonometric functions are involved I am not able to solve it. Please help in finding it by hand.
Dharmendra Singh
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Prove range for derivative given $3$ definite integrals

Let $f:[0,1]\longrightarrow \Bbb R$ be a continuously differentiable function such that $$\int\limits_{0}^{1}f(x)dx=1,\int\limits_{0}^{1}xf(x)dx=2,\int\limits_{0}^{1}x^2f(x)dx=3$$ Prove that for every $t\in[-24,60]$, there exists $c\in(0,1)$ such…
math110
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Integrating :$\int\sqrt{\sin x} \cos^{\frac{3}{2}}x dx$

How to integrate : $$\int\sqrt{\sin x} \cos^{\frac{3}{2}}x dx$$
sax
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A trigonometric integral (guessed from a combinatorics formula)

In class, I defined the binomial coefficient using an integral: $$\binom{n}{k} = \displaystyle \int_0^{2\pi}\dfrac{dt}{2\pi} e^{-ikt}(1+e^{it})^n.$$ I succeeded in demonstrating many standard properties of the binomial coefficient directly using…
Isomorphism
  • 5,693
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How to show that $\int_{0}^{\infty}{\sin(x)\sin(2x)\sin(4x)\cdots\sin(2^kx)\over x^{k+1}}\mathrm dx=2^{0.5(k^2-k-2)}\pi?$

$$\int_{0}^{\infty}{\sin(x)\sin(2x)\sin(4x)\cdots\sin(2^kx)\over x^{k+1}}\mathrm dx=2^{0.5(k^2-k-2)}\pi\tag1$$ $k\ge0$ Experimental using wolfram integrator, let me to conclude the closed form for $(1)$ is as follow above. But I don't know how shows…
gymbvghjkgkjkhgfkl
  • 8,090
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Evaluating $\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta$

I need solve this integral, and I tried various methods of solving and did not get it. The integral is: $$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$$ where $t$ is a positive integer.
Patrícia Tempesta
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