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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.

20559 questions
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number of such real values of $a.$

Let $a$ be a real number in the interval $[0,314]$ such that $$\displaystyle \int^{3\pi+a}_{\pi+a}|x-a-\pi|\sin \frac{x}{2}dx=-16.$$ Determine the number of such real values of $a.$ What I tried: Put $x-a-\pi=t$. Then, $$\displaystyle…
jacky
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Fundamental theorem of calculus part 2

In the fundamental theorem of calculus part 2 we have $$\int_a^bf(x)dx=F(b)-F(a)$$ where $F(x)$ is any anti derivative of $f(x)$ on $[a, b]$ but i want to know that what will happen if the equality $F'(x)=f(x)$ does not hold at some point in $[a,…
Pranava Priyanshu
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$\int_{0}^{\pi/2} (\sin x)^{1+\sqrt2} dx$ and $\int_{0}^{\pi/2} (\sin x)^{\sqrt2\space-1} dx $

How do I evaluate $$\int_{0}^{\pi/2} (\sin x)^{1+\sqrt2} dx\quad \text{ and }\quad \int_{0}^{\pi/2} (\sin x)^{\sqrt2\space-1} dx \quad ?$$
Souvik Dey
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Differentiation under the integral sign? $\int_0^1 (x\ln(x))^{50} \mathrm{d}x$

$$\int_0^1 (x\ln(x))^{50}\,dx$$ This was listed as a question using Differentiation Under the Integral Sign. How do I solve this? First, I tried introducing a parameter $p$. $$I(p)=\int_0^1 (x\ln(x))^{p}\,dx$$ Then, I differentiated it with…
helpme
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Definite Integral $\int_0^2 \frac{\ln(1+x)}{x^2-x+1}dx$

I was working on this problem, I probably need to abuse symmetry somewhere but can't see how: $$\int_0^2 \frac{\ln(1+x)}{x^2-x+1}dx$$
mtheorylord
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Evaluate $\int\limits ^{0}_{-\infty }\frac{\ln( t+1)}{t^{2} +1} dt$

I have been trying to solve this integral for some time now $\int\limits ^{0}_{-\infty }\frac{\ln( t+1)}{t^{2} +1} dt$. and all the calculators I've used say it's equal to $\frac\pi4\ln(2)-G+\frac{π^{2}i}{4}$ ($G$ is Catalan's Constant), but I find…
Spador Yedi
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Setting up the triple integral in spherical coordinates in a different order

Let D be the region bounded below by the plane $z=0$, above by the sphere $x^2+y^2+z^2 = 4$ and on the sides by the cylinder $x^2+y^2 = 1$ Set up the triple integral in spherical coordinates that give the volume of D using the following order of…
Archer
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Integrating and Triangle after Transformation to a standard simplex

I am new to this forum and even don't know how to write these mathematical symbols. I have a question below and your reply would be highly appreciated. If I have a triangle with vertices $V_1 = (2,2)$, $V_2 = (4,2)$, $V_3 = (2,4)$. I can find its…
assadabbasi
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What value of limits should I substitute in the arcsin(x/a) term in the formula for ∫(4-x²)½ dx where x goes from -1 to 1

Logically the area would be half a circle=2π Ok so, we have this formula. The only problem is the arcsin term gives π/6,5π/6,-7π/6,-11π/6 and their nπ multiples for x=1 in the formula. A similar situation holds for the arcsin term with x=-1 Which…
Nigel Goveas
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Eval definite integral power of sine via seperate interval

The question is $\int_0^{\pi/2} \sin^n x \ \mathrm{d}x$. We know that this can be solve with a reduction formula if n is definite. And easily guess it results to 0 if $ n \rightarrow +\infty $. This article says there is another way: $\int_0^{\pi/2}…
cuter44
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Doubt in an example for the Definite Integral chapter of Demidovich's book

So the example is in the page 139 of Chapter V, Definite Integrals Form the integral sum $S_n$ for the function $$ f(x) = 1 + x $$ on the interval $[1,10]$ by dividing the interval into n equal parts and choosing points $\xi_i $ that coincide with…
highentropy1915
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Integral $\int_0^b \frac{1-\cos ax}{x} dx$

How can I evaluate this? $$ I=\int_0^b \frac{1-\cos ax}{x} dx$$ I tried the following two approaches that do not work well: Use the power series expansion $$ \frac{1-\cos ax}{x}=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n)!} a^{2n}x^{2n-1}$$ and…
Laplacian
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How to solve the given definite integral?

This integral appears on solving a particular problem of gamma ray detectors. I have tried several substitutions and methods but can't solve it. Maybe it uses some special functions. $$\int_{\theta_1}^{\theta_2} \cos^{-1}(a\hspace{0.05cm}\cot…
Asit Srivastava
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A plane figure is bounded by the curves $2y=x^2$ and $x^3y=16$...

A plane figure is bounded by the curves $2y=x^2$ and $x^3y=16$, the x-axis and the ordinate at x=4. Calculate the area enclosed. My attempt: $2y=x^2,\; x^3y=16 \\ y=\frac{x^2}{2} \quad y= \frac{16}{x^3}\\$ $\int ^?_?\biggr[\frac{x^2}{2}…
Orestes Dante
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Area bounded by a curve and a straight line

Prove that the area bounded by the curve $y=\tanh x$ and the straight line $y=1$, between $x=0$ and $x=\infty$ is $\ln 2$ $\int^\infty _0 (1-\tanh x) \mathrm{dx}= \biggr[x-\ln (\cosh x)\biggr]^\infty_0\\\infty - \ln(\cosh \infty)+1$ How do I get…
Orestes Dante
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