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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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Calculus definite integration question

I was doing the question $$\int_{-2}^{1} \{x\} dx =\frac{3}{2}$$ which got me wondering for the general expression and after few trials i got to the expression for $$\int_{a}^{b} \{nx\} dx = \left(\frac{[nb]-[na]}{2n}\right) +…
Neelamber Mishra
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Evaluation of certain definite integrals

How do we integrate the following two integrals? $$ \int_{-e}^\pi \cos(-3x^2) \, dx$$ and $$ \int_{-e}^\pi \frac {e^x}{\ln(a-x)} \, dx,$$ where $a>\pi$.
Souvik Dey
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Arranging $\int^{\pi/2}_0\sin(\cos x)dx$, $\int^{\pi/2}_0\cos(\sin x)dx$, $\int^{\pi/2}_0\cos(x)dx$ in increasing order

Arrange the following in increasing order $$ A =\int^{\pi/2}_0\sin(\cos x)dx \qquad B =\int^{\pi/2}_0\cos(\sin x)dx \qquad C =\int^{\pi/2}_0\cos(x)dx $$ What I try as We know that $\displaystyle…
jacky
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Integration of Chebyshev Polynomials

I have a task of expanding a Lorentzian representation of a Dirac - Delta with Chebyhsev polynomials, basically, the integral I need to perform is $$\frac{1}{\pi}\int_{-1}^{1}dx\frac{\eta}{(x-\mu)^2 + \eta ^2} \cos (n\cdot \arccos(x))$$ where $n$ is…
darkness101
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How to evaluate the following integral?$\int_1^2\frac{\sqrt{x}}{1+x+x^2+x^3}$

$$\int_1^2\frac{\sqrt{x}}{1+x+x^2+x^3}$$ How to evaluate the following integral? I am trying this question by simplifying the denominator. The denominator will be $(1+x)(1+x²)$ and then I am thinking for applying partial fraction method to solve the…
DEB SANKAR ROY
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Find area under inverse of a cubic function.

$ N = 2^a × 3^b × 5^c $ And the fractional part of $ \sqrt{N} = 0 $ a ,b, c (natural nos.) are the roots of cubic $ f(x) = x³ - px² + qx -8 $ Find out the area bound by the curve $ f ^{-1}(x) $ , y = 0 and x= 8 . I found out that N must be a…
Arham Kothari
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Volume of a cissoid?

A cissoid with formula $y^2(2a - x) = x^3$ where $a>0$ How do I show the the volume of the cissoid rotated about the asymptote is: $$V = 2\pi \int_0^{2a} (3a - x)(2ax - x^2)^{1/2} dx$$
Eric
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Evaluate $\int_{0}^{0.5} e^{-0.5\frac{a}{2u-1}} (\frac{u}{1-u})^{2b-1} du$

Is there a closed form expression( special functions are allowed) for the following integral: $$J(a,b)=\int_{0}^{0.5} e^{-0.5\frac{a}{2u-1}} \left(\frac{u}{1-u} \right)^{2b-1} \,du$$ where $a<0$ and $b\in (0,1)$. Simulation studies showed that…
Masoud
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Finding all solutions $f$ of $\int_0^{\pi/2} f(\theta_i, \theta_o) \sin 2\theta_o d\theta_o=\sin^2\theta_i$ with conditions $f(a, b) = f(b,a) \geq 0$

My problem is the following: I am looking for all possible functions $f(\theta_\text{i}, \theta_\text{o})$ with $f(\theta_\text{i}, \theta_\text{o}) \in \mathbb{R} \ \forall \ (\theta_\text{i}, \theta_\text{o}) \in [0, \pi/2]^2$ that verify the…
Balfar
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Prove that $\:I_{n+2}+I_n=2I_{n+1}\,,\;$ given $\;I_{n}=\int^{\pi}_0\frac{4-\cos(n-1)x-2\cos(nx)-\cos(n+1)x}{1-\cos x}\mathrm dx$.

Let $\displaystyle I_{n}\!=\!\int^{\pi}_0\frac{4-\cos(n\!-\!1)x-2\cos(nx)-\cos(n\!+\!1)x}{1-\cos x}\,\mathrm dx\\[4pt]$ Then prove that $\displaystyle I_{n+2}+I_n=2I_{n+1}$ My Try : Using $\displaystyle\;\cos C+\cos D…
juantheron
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Solving the integral $ \int_{0}^{\infty} e^{-x}\cos(x) \, dx $

Problem. Calculate $$ \int_{0}^{\infty} e^{-x}\cos(x) \, dx $$ I was recommended to calculate it using limits, but firstly I have solved the integral and I got $$ \int_{0}^{\infty} e^{-x}\cos(x) \, dx = \frac{\sin(x) - \cos(x)}{2e^x}. $$ I don't…
Sofia Ibragimova
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find the value of ${\int_{0}}^1 (1+e^{-x^2})dx$

How to find $${\int_{0}}^1 (1+e^{-x^2})dx$$ How do I integrate $e^{-x^2}$ it can't be done by parts.
DebojyotiMukh
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Find the upper bound of definite integral using the result

I am making a script for kerbal space program, which will calculate the height where a "hover slam" should be started when the rocket is full. I have worked out that the velocity after a certain amount of time is: $Δv=\int^t_{0}{\frac{T}{m_0*Bt}}…
an inconspicuous semicolon
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How to compute a definite integral of the form $\int_a^b f(x)g(x)dx$? Is there perhaps a trapezoidal sum formula for $f(x)*g(x)$?

I need help with this to verify results that I obtained while running a program in Matlab which utilized its built in Trapezoidal Sum function to numerically compute an integral. So, here is an example that I went over which follows the form of…
Researcher R
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Integration of $\int_0^{1/4} \frac{x^3}{ \sqrt{1-9x^2}} dx$ using trigo substitution

Evalute $\int_0^{1/4} \frac{x^3}{ \sqrt{1-9x^2}} dx$ using trigo substitution Because $\sqrt{a^2 - b^2x^2} => x = a/b \sin \theta$ So, Let $x= \frac{1}{3} \sin \theta$ $dx = \frac{1}{3} \cos \theta$ Elimination of roots: $\sqrt{1-9x^2} = \sqrt{1-…
user307640
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