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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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Finding $ \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\bigg(\frac{1+x}{1-x}\bigg)dx$

Finding $\displaystyle \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\bigg(\frac{1+x}{1-x}\bigg)dx$ Try: Let $$\displaystyle I = \int^{1}_{-1}\frac{x^3}{\sqrt{1-x^2}}\ln\bigg(\frac{1+x}{1-x}\bigg)dx$$ (Function is even ) $$I =…
DXT
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Evaluating an integral without using the Digamma function or complex analysis.

Suppose that $n$ is a positive integer, and let $k$ be a positive integer less than $n$. How can one show that $\displaystyle\int_0^1 \frac{x^{k-1} - x^{n-k-1}}{1 - x^n} \; dx= \frac{\pi}{n} \cot{\frac{\pi k}{n}}$ without using the digamma (psi)…
BDS
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If $f(x)=\int_1^x\frac{\tan^{-1}t}t\,\mathrm dt$, find $f(e^2)-f(\frac1{e^2})$

If $$f(x)=\int_1^x\frac{\tan^{-1}t}{t}\,\mathrm dt,$$ find $f(e^2)-f\left(\dfrac{1}{e^2}\right)$. I think that Newton-Leibniz formula should be applied but the problem is that that if I apply it, I would get $f'(x)$ but I have to find the answer…
Soham
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Evaluate $\int_0^\pi \frac{\sin^2 nx}{\sin^2 x}dx$

How do I evaluate this definite integral, I'm not even getting a slightest idea on how to approach this. Tried converting into cos using double angle property but that didn't help. $$\int_0^\pi \frac{\sin^2 nx}{\sin^2 x}dx$$
Aryan Verma
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Integration of an inverse function

I need to evaluate $$\int_0^1\frac{\sin^{-1}x}x dx.$$ I have tried integration by parts (taking $1/x$ as second function) but I am getting complex results. Any idea on how to approach will be appreciated.
user562886
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Can this integral similar to $\int_{-\infty}^{\infty} \frac{\sin(x)}{x} dx$ be evaluated?

I know that the integral $\int_{-\infty}^{\infty} \frac{\sin(x)}{x}dx = \pi$. Just for fun, I thought if it would be possible evaluate the same integral but only consider the area below the curve and above the x-axis? Approximately would be fine,…
Adam
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On the integral $ \int \cos(x)^{{\cos(x)}}\mathrm{d}x$.

I would like to know if there is a closed formula for the integral $$ \int \cos(x)^{{\cos(x)}}\mathrm{d}x\overset{? }{=} $$ in terms of $\pm x^{\pm n}$, $(\pm\cos(x))^{\pm 1}$, $(\pm\sin(x))^{\pm 1}$, $ e^{\pm x}$, $ e^{\pm\cos(x)}$,$…
Elias Costa
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Alternative computation of $\int_0^1 \frac{\ln^2(1-x^4)}{1+x^2}dx$

I know how to prove that, $\displaystyle K=\int_0^1 \frac{\ln^2(1-x^4)}{1+x^2}\,dx=\frac{9}{4}\pi\ln^2 2+\frac{7}{48}\pi^3-12G\ln 2$ (see: Closed form for the following integrals $ 1)\int_0^1\frac{\ln(1-x^2)\ln (1+x^2)}{1+x^2} \; dx$ ) But, i'm…
FDP
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The number of continuous functions $f:[0,1]\to\mathbb R$ that satisfy $\int_0^1xf(x)\,dx=\frac13+\frac14\int_0^1(f(x))^2\,dx$

90) The number of continuous functions $f:[0,1]\to\mathbb R$ that satisfy $$\int_0^1xf(x)\,dx=\frac13+\frac14\int_0^1(f(x))^2\,dx$$ is A) 0 B) 1 C) 2 D) $\infty$ How to approach this sum? I thought of using Newton-Leibniz but the…
Alphanerd
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Hypervolume under the square of an n-simplex

I posted this question a while ago, but since I didn't have much luck I though I'd reformulate it and try again. Question: What is the general form of the equation that gives the hypervolume under the square of an $n$-simplex in an $n+1$ dimensional…
edgar
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How to evaluate the integral $\int \limits_0^\infty \frac{\sin x}{x^{1/3}}\,\mathrm{d}x$?

Is there any special convergence factor like $\frac{1}{x}\to\int \limits_0^\infty e^{-xy}\mathrm{d}y$ that can transform the $\frac{1}{x^{1/3}}$? I don't mean $\int \limits_0^\infty e^{-x^{1/3}y}\mathrm{d}y$.
Tongho
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Integration with respect to a parameter $a$

Show that the value of the integral of $$\frac{\log(1+\cos(a)\cos(x))}{\cos(x)}$$ over the interval $[0,\pi]$ is given by $$\pi\left(\frac{\pi}{2} - a\right)$$ for $0\leq\,a\leq\,2.$
Callie12
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Evaluating $\int_{0}^{x}\frac{\arctan(t)}{t}dt-\int_{0}^{\frac{1}{x}}\frac{\arctan(t)}{t}dt$

The paper here https://www.imsc.res.in/~rao/ramanujan/collectedpapers/Integral/Integral1.htm says that, $\int_{0}^{x}\frac{\arctan(t)}{t}dt-\int_{0}^{\frac{1}{x}}\frac{\arctan(t)}{t}dt=\frac{1}{2}\pi\ln(x)$ I tried verifying this result graphically…
Mathoholic99
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Doubt in a Definite integral Concept

$$ I = \int_0^{4\pi} \frac{\sec^2x\ dx}{2+\tan^2x} $$ Now it is simple and I can just write it as, $I=\left[\frac{1}{√2}\tan^{-1}(\tan x/√2)\right]_0^{4\pi}$. Now if I simply place limits I will get $0-0=0$. But if I take $8$ common as…
TheLostGuardian0
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Proof that sum of integrals of same function on adjacent intervals equals the integral over both intervals

I am seeking a proof that $$\int_a^c f(x) dx + \int_c^b f(x) dx = \int_a^b f(x) dx$$ Please use Stewart's (Calculus: Early Transcendentals, 2016, 8e, p. 378) definition of the definite integral: This is what I have: Suppose $$u=\int_a^c f(x) dx =…
pnn001
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