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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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Parameter Integral Sin function, Gamma function

Given $$ F: A \in\Bbb{R}\rightarrow\Bbb{R}$$ such that $$F(y)=\int\limits_{0}^{\pi/2} (sinx)^y(cosx)^{1-y} \; dx$$ Prove that $F(1/2)=\frac{1}{\sqrt\pi}(\Gamma(3/4))^2$ and then find the maximum domain $A$ such that $F$ exists. $$\Gamma(s)=…
prometheus21
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How is the integral $\int \frac{1}{ae^{mx}+be^{-mx}} \, dx$ solved?

I am trying to determine how the following integral was solved, or at least the name of article deriving this solution: $$ \int \frac{1}{ae^{mx}+be^{-mx}} \, dx = \frac{1}{m\sqrt{ab}}\,\tan^{-1} \left(e^{mx}\sqrt{\frac{a}{b}}\,\right)$$
Aschoolar
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Stokes' theorem to evaluate an integral

Let $C$ be the intersection curve of the parabolic sheet $y=x^2$ with the cylinder $x^2+z^2=4$, oriented clockwise when viewed from the positive $y$-axis. Apply Stoke's Theorem to the integral $$ \int_C 2y\,dx+xz\,dy+z^2\,dz$$ and continue until…
Diamom
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Define $\lim_{x\rightarrow 0} \frac{1}{x}\int_{0}^{x} e^{t^{2}} dt$, what is the purpose of this question?

This question is in the section about definite integrals and the task is to calculate the limit. My first idea was division-by-zero but I am very unsure about this. What is the goal here? I then thought that should I investigate things by different…
hhh
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Prove that $\int_0^{2\pi}{\frac {\sin^2\theta (cos\theta - a)}{(1-a \,cos \theta)^4}}d \theta = 0$ for $0<=a<1$

I am trying to prove that for $0<=a<1$ $$\int_0^{2\pi}{\frac {\sin^2\theta (cos\theta - a)}{(1-a \,cos \theta)^4}}d \theta = 0$$ I know that $$\int_0^{2\pi}{\frac {\sin\theta \,cos\theta}{(1-a \,cos \theta)^4}}d \theta = 0$$ and…
steveOw
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find area of enmclosed b these curves$ f(x) = x^3 $and $g(x) = x^5 -2x^3-3x$

i have calculated this using "Mathematica" but seems like i a getting negative answer here is what I did Solve[x^5 - 3 x^3 - 3 x == 0, x]. I calculated the value for x and then i integrated the 2 equations from 0 to square root[1/2 (3 + square…
shane
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Verify (Riemann) integrability proposition with counterexamples

Let $f[a,b]: \mathbb R \rightarrow\mathbb R$ be a (Riemann) integrable function such that $f \geq0$ and $\int_{a}^{b}f = 0$. Verify with counterexamples that the sentence aforementioned conditions do not imply $f\equiv 0$. I couldn't think of any…
user71487
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Explaining why $\int_{b-\frac{1}{n}}^{b-\frac{1}{2n}}\text{affin}=\text {area of triangle}$

Little earlier laid out an example of which can be found at: Explaining why $\Vert x_n\Vert _1=\frac{3}{4}$ I do not understand why: $$\int_{b-\frac{1}{n}}^{b-\frac{1}{2n}}\text{affin}=\text {area of triangle}$$ I hope someone will help me…
user145717
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Length travelled of a sine wave at given amplitude and frequency

I recently read the following post: What is the length of a sine wave from $0$ to $2\pi$? This covers the equation y = sin(x) with no particular units. To describe any sine wave, we use the equation y = Asin(wx + phi) where A is the amplitude, w is…
Trashman
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Definite Integral Evaluation

Evaluate $\displaystyle \int_{0}^{2}x^3\sqrt{(2x-x^2)} dx$ This kind of problem is solved using tricks like putting $\displaystyle x=\sin^2t$ and/or identities like $$\begin{align} \int_{0}^{a}f(x) dx &=\int_{0}^{a}f(a-x)…
square_one
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Order of New Limits of Integration

$$\int_{\pi/4}^{\pi/2} \cot^5 \phi \csc^3 \phi \;d\phi$$ My final answer is: $$\frac{8}{105} - \sqrt 2 \left(\frac{22}{105}\right)$$ This is apparently the wrong answer. Did I make a mistake somewhere ... I made the substitution let $u =…
Guest
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Two different results for the evaluation of a specific definite integral

Hi I am unsure of my attempt to solve the following: $$ \int_{-1}^1 xde^{|x|} $$ my attempt is the following: Using substitution where $ u = x, v = ?, dv = e^{|x|} $ Now : $$ v = \int_{-1}^1 dv = \int_{-1}^1 e^{|x|}dx$$ $$= \int_{-1}^0 e^{-x}dx +…
user146387
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Computing two integrals - per-partes?

I start with integrals and attempting to figure out these two integrals, but can't move from a spot $\int x^2 lnx dx$ $\int \frac{lnx}{\sqrt[3]{x}}dx$ The first example - it doesn't look so complicated, but I just can't get the right result. And…
user984621
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Integral $\int_{0}^{\pi}\sqrt{1+4\sin^{2}(x/2)-4\sin(x/2)}\mathrm{d}x$

Here's how I solved it. \begin{eqnarray*} & & \int_{0}^{\pi}\sqrt{1+\left(4\sin^{2}\left(\frac{x}{2}\right)\right)-\left(4\sin\left(\frac{x}{2}\right)\right)}\mathrm{d}x\\ & = &…
user80551
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An integral related to the power and algebraic function

I meet with a complex integral problem, given as follws: $$\int_0^\infty {\frac{{{x^p}}}{{{{\left( {x + a} \right)}^q}}} \cdot {{\left( {\frac{{x + b}}{{x + c}}} \right)}^n}dx,where{\text{ }}q > p + \frac{3}{2};a,b,c > 0;n = 0,1,2,3...} ,\infty $$
Chaoqing Tang
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