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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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Integrating $\int_{0}^{\pi} \frac{x(\sin x)^6}{(\sin x)^6+ (\cos x)^6}\,\mathrm dx$

Can't really wrap my head around this integral. I tried doing some u-subs, but it just doesn't get much easier. Would really appreciate any help. $$\int_{0}^{\pi} \frac{x(\sin x)^6}{(\sin x)^6+ (\cos x)^6}\,\mathrm dx$$
David Dragan
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solve $\int_1^x \frac{1}{t^{2023}-t^{2022}+t^{2021}-t^{2020}} dt$

Solve the rational integral $$\int_1^x \frac{1}{t^{2023}-t^{2022}+t^{2021}-t^{2020}} dt.$$ I am trying to solve a IVP and this is the last step to solving it, but I can't get through it. I tried the ordinary method of trying to separate them into…
Csaba Daniel Farkaš
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How to compute the integral of cos^2(x)+a under square root?

I would need to compute the following integral: $$ \int_0^{2\pi} \sqrt{\cos^2(x)+a}\ dx$$ with $a$ real and positive. Is there any closed analytical expression for this integral? Thank you very much for your attention. Best Regards.
Marco Gandolfi
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Proving $\int_{0}^{\infty} e^{-ax}\sin^{p}(x) dx$

My text book says that $$ I = \int_{0}^\infty e^{-ax} sin^p ~x ~dx = \frac {p (p – 1)}{p^2 + a^2} \int_{0}^\infty e^{-ax} sin^{p-2}~x~dx $$ Can someone help me find this result? Because I only see when differentiating by parts $$ u = sin^p (x)…
velsd
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Give an example of a function that is integrable on the interval [0, 1] and incomplete on [0, 2].

Give an example of a function that is integrable on the interval [0, 1] and not-integrable on [0, 2]. So , i was thinking about function ={-1, 0<=x<=1 and 0, 1
Karol Bargieł
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The $ \int_Tf(x)g(y)dxdy$ equals?

Given two real-valued functions f(x) and g(x) both defined by definition x if x $\in [0,1] $, 0 otherwise. Let T denote the triangle in the real plane bounded by x axis, y axis and line x+y=2. The $ \int_Tf(x)g(y)dxdy$ equals? I think it should…
Vikas Sharma
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Setting up a double integral for the region $R$ bounded by $y=3x^2-1, y=x^2, y=0$.

I am practising setting up limits of integration for double integrals. I'm struggling with the question below, so I was hoping to get some feedback on my current solution (for the limit portion anyway). I have been tasked with integrating the…
Pepe Silvia
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Stuck on a definite integral using trigonometric substitution

I got stuck solving this definite integral using trigonometric substitution $$F(x)=\int_{-\sqrt{r^{2}-x^{2}}}^{\sqrt{r^{2}-x^{2}}}\sqrt{r^{2}-(x^{2}+y^{2})}dy$$ where $-r\leqslant x\leqslant r$ I let $$y=r\sin{\theta}$$ $$dy=r\cos{\theta}…
kauselis3000
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If $f(x) + f(x + 1) = 2$, $I_1 = \int^8_0 f(x)dx$ and $I_2 = \int^3_{-1} f(x) dx$ then find $I_1 + 2I_2$.

Given $f(x) + f(x + 1) = 2$ (f is continuous) $I_1 = \int^8_0 f(x)dx$ $I_2 = \int^3_{-1} f(x) dx$ Find $I_1 + 2I_2$. I found $f(x) = f(x+2)$ so function has a period of 2. So $I_1 = 4\int^2_0 f(x)dx$ but I'm not getting how to solve this.
Shub
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How to calculate $\int_0^\infty e^{-x^2}\mathrm{d}x$ given the following conditions?

The definite integral $\int_0^\infty e^{-x^2}\mathrm{d}x=\frac{\sqrt{\pi}}{2}$ is very typical. We have: $$1-x^2 \leqslant e^{-x^2}(0 \leqslant x \leqslant 1), \quad e^{-x^2} \leqslant \frac{1}{1+x^2} \quad (x \geqslant 0)$$ Besides, the…
ZixuanLing
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Find my error in evaluation of $\int_0^{\infty} \frac{\cos x}{e^x+e^{-x}}$?

I was trying to evaluate $$I=\int_0^{\infty} \frac{\cos x}{e^x+e^{-x}}\text{d}x,$$ and am trying to find my error. Here's my work, with tons of algebra left…
Moni145
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Integrable in 3 variables

I have to compute $$\iiint_D(x^2+2y^2)z\, dxdydz$$ with $D=\{x^2+y^2+z^2\leq 4, z>\sqrt{x^2+y^2}\}$. I tried with some classical change of variables but without more sense. Same ideas?
Mario
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Finding $\int_0^{2\pi}\,d\theta\,\,\frac{a-i\cdot{b}\cos(\pi+\theta)}{c-i\cdot{d}\cos(\pi+\theta)}$

I would like to calculate the following definite integral: $$\int_0^{2\pi}\,d\theta\,\,\frac{a-i\cdot{b}\cos(\pi+\theta)}{c-i\cdot{d}\cos(\pi+\theta)}$$ Anybody has any suggestion? thanks So according to the hints, I should proceed in this…
JFNJr
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Boundedness of $\int_0^t e^{-(t-\tau)}x(\tau) d\tau$

I want to bound the following function $y$. Given a differentiable and monotone decreasing function $x(\tau)$, define $y$ as follows: $$y(t)= \int_0^t e^{-(t-\tau)} x(\tau) d\tau.$$ While searching about the function, there is no result about…
J Light
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Closed form of $\displaystyle \int_{1}^{x}\cos\left(\pi\left(n+1\right)\right)e^{\frac{x}{n}}dn$

The graph of $$ \int_{1}^{x}\cos\left(\pi\left(n+1\right)\right)e^{\frac{x}{n}}dn $$ looks really similar to $\Gamma(x)$ plus some sinusoidal function. Wolfram Alpha couldn't give an answer. Is there some way to evaluate this integral? Maybe in…
Michael Smith
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