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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Definite integral of [{Cos^-1(x)}]/√(1-x^2) from 0 to 1

If we substitute $cos^{-1}(x)$ with $z$ then,corresponding values of $x=1$ are $z=0,2\pi,4\pi...$ And correspond values of $x=0$ are $z=\pi/2,3\pi/2,5\pi/2...$ Now if for $x$ belongs to $[0,1]$ corresponding value is taken $z$ belongs to $[\pi/2,0]$…
MSKB
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definite integral $\int_{-1}^1x^2\sqrt{4-x^2}dx$

I have solved the problem and obtained the value of the integral $2\pi/3-\sqrt{3}/2$. In the actual solution of this problem I have seen that $x$ was substituted by $2\sin z$ and upper limit $1$ was substituted by $\pi/6$ and lower limit $-1$ by…
MSKB
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Calculating the integral of $\int_0^\infty \int _{-\infty}^\infty \frac{x e^{-x}}{0.2(\sin(1.5x)+y)^2+0.1}dy dx$

I was wondering if this has a name (I was taking the Gamma distribution and tried to create a 2D with "wiggly-tail" version of it). And also if there is a way to solve it? I used numerical methods to estimate that the integral (i.e.…
Maverick Meerkat
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Help regarding the limit of an integral

The function $g(t)$ givenby $$ g(t)=\frac{n(n-1)}{2}\left[\frac{t^{n-2}(2-t)^{n+1}-t^{2n-1}}{n+1}-\frac{t^{n}(2-t)^{n-1}-t^{2n-1}}{n-1}\right]$$ integrates to 1 form $t=0$ to $t=1$.From numerical simulations one can see that limit of $\int_0^1…
AgnostMystic
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u-substitution of definite integral

$$\int_1^3 \frac{(\ln(v+1))^2}{(v+1)}dv$$ Here is what I have done: $u = v+1$ $\frac{du}{dx}=1$ $du = dv$ $$\int_2^4 \frac{(\ln(u))^2}{(u)}du$$ Then $z=\ln(u)$ $\frac{dz}{du}=\frac{1}{u}$ $dz = \frac{du}{u}$ $$\int_2^4 \frac{(z)^2}{(u)^2}dz$$ And…
SI Ma
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Integrating vertically giving different answer from intergrating horizontal

Find area in the region: $$ \frac{1}{x}
tryst with freedom
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dividing integrals to derive the equations

I'm trying to derive an equation for which I have a relationship between a constant and its integral -- namely, $$W = 2 \int_0^\infty f(L)\,dL$$ -- but also have another definition for $W$ -- $$W = -L\ln(1-P)$$ $f(L)$ itself is $f(L) = P_d$, a value…
Ken Chiacchia
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Two definite integrals in terms of one another

While practicing definite integration I stumbled upon following question: Q.) If $\int_{0}^{π/2}\ln(\sin x)dx=k$ then, find $\int_{0}^{π/2}\frac{x²}{(\sin x)²}dx$, in terms of $k$ My attempt: I have memorised the first integral answer…
Kshitij Kumar
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how to solve $\int_{0}^{1}\int_{y}^{1} e^{x^2}dxdy$?

$$\int_{0}^{1}\int_{y}^{1} e^{x^2}dxdy$$ I tried this this: $$\int_{y}^{1} e^{x^2}dx=\sum_{0}^{\infty}\frac{1}{n!}(\frac{1}{2n+1}-\frac{y^{2n+1}}{2n+1})$$ I don't know what to do after or even if this way works...
Elsa Hejazian
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Polar coordinate limits for Definite integral of $\int_{a}^{\infty}t^2e^{t^2}$ dt

Can anyone help me with this integral $$I = \int_{a}^{\infty}t^2e^{-t^2} dt$$ $$I^2 = \int_{a}^{\infty}\int_{a}^{\infty}t^2h^2e^{-t^2-h^2} dt dh$$ Can you please explain how I can find the limits in the polar coordinates for the above integral? I am…
Unni
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Definite Integral for $\int_{0}^{a} x^{−n}dx$

Definite Integral for $\int_{0}^{a} x^{−n}dx$ where $n<=0$ and $0 < x < a$ $\int x^{−n}dx = x^{(-n + 1)}/(-n + 1)$. I'm not able to plug in the lower limit $0$ here because won't the result reaches $\infty$? I'm sure something is wrong with the…
SrikRaja
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Calculate the integral $\int_{0}^{1+\log \sqrt 3} |\frac 32 e^{1-x} -\frac 12 e^{x-1}|$

This is small part of a much larger question. My trouble is calculating with the modulus function on. Since I don’t really know which function is greater than the other at, I can’t tell how it will open up. Assuming that there is no graphing…
Aditya
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Integration using Incomplete Gamma function

Evaluate ${I_1}\left( z \right) = \int\limits_0^z {{{\left( {1 - x} \right)}^{b - 1}}{e^{ - ax}}} dx;\,a,b > 0\,,0 < z < 1$ I have solved using incomplete Gamma function. By a substitution $\left( {x - 1} \right)a = t$, the above integration…
user827039
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If $I=\int_{0}^{\frac 12} \frac{1}{\sqrt{1-x^{2n}}} dx$, then prove that $I<1$ for $n\ge 1$

The maximum value of $\frac{1}{\sqrt{1-x^{2n}}}$ is $\frac{2^n}{\sqrt{2^{2n}-1}}$ So the maximum value for is the integral is $$\frac 12 \frac{2^n}{\sqrt{2^{2n}-1}}$$ Now I don’t know how to find the max value from here. I tried to use $\lim_{n\to…
Aditya
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please help me getting the correct answer to following problem of integral calculus

Let f=n $(\sin x)^{2n+1}\cos x$ then the value of $$\lim_{n\to \infty}\int_0^{\pi/2}f dx -\int_0^{\pi/2}(\lim_{n\to \infty}f)dx$$ is (a)1/2 (b)0 (c)-1/2 (d)-$\infty$ the correct answer is (a) but i dont understand how. I evaluated $\lim_{n\to…
Shivam Mishra
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