# Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

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### Integral of $(T+dt)^{4}$

I want to take the anti derivative of $\frac{dq}{dt}=\beta \Delta T - (T+ \Delta T)^{4}$. I'm not sure how to go at it. I want to put it as $\frac{dq}{dt}=\beta dT - (T+ dT)^{4}$. I am worried about ...
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### Prove that $\int_0^1 f(x)g(x)\,dx \ge \int_0^1 f(x)\,dx \cdot \int_0^1 g(x)\,dx$

How to prove that if $f$ and $g$ are continuous and increase monotonically on $[0,1]$, then $$\int_0^1 f(x)g(x)\,dx \ge \int_0^1 f(x)\,dx \cdot \int_0^1 g(x)\,dx \quad ?$$ The integral is understood ...
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### Evaluate $\int_{0}^{\frac{\pi}{4}} \sin\left(\arccos{\left(\frac{\cos{x + 5}}{1 + 2\cos{x}}\right)}\right) \,dx$

Evaluate the following integral $$\int_{0}^{\frac{\pi}{4}} \sin\left(\arccos{\left(\frac{\cos{x + 5}}{1 + 2\cos{x}}\right)}\right) \,dx$$ Working: We get rid of the $\sin$ by considering the triangle. ...
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Evaluate $$\int_0^\pi\mathrm{e}^{\left|\cos\left(x\right)\right|}\sin\left(x\right)\left(2\sin\left(\dfrac{\cos\left(x\right)}{2}\right)+3\cos\left(\dfrac{\cos\left(x\right)}{2}\right)\right)\mathrm{d}... 2 votes 1 answer 103 views ### Evaluating \lim_{\lambda \rightarrow 0}\bigg(\int_0^1 (\beta x + \alpha(1-x))^\lambda dx\bigg)^{1/\lambda} Given 0 < \alpha < \beta, I want to evaluate the limit of the integral$$\lim_{\lambda \rightarrow 0}\bigg(\int_0^1 (\beta x + \alpha(1-x))^\lambda dx\bigg)^{1/\lambda}$$I attempted u-... • 2,006 2 votes 0 answers 21 views ### Possibility of computing antiderivative using dual numbers It is known that, given a function f(x), plugging in the dual number x+\varepsilon, where \varepsilon^2=0, yields f(x) + f'(x)\varepsilon. For example:$$f(x) = x^3\\f(x+\varepsilon)=(x+\...
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How to integrate the following integral, which I have no idea. Thanks for your attention. $$I=\int_0^{2\pi}d \theta \int_0^{\pi} e^{\sin\varphi(\cos\theta -\sin\theta)} \sin\varphi\, d \varphi$$ I ...