# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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### Evaluate the improper integral $\int_0^\infty x^{-1/2}e^{-(x-a)^2}dx$

Here is an integral I encountered in the wild: $$\int_0^\infty x^{-1/2}e^{-(x-a)^2}dx .$$ If we substitute $t = x^{1/2}, dt = \frac{1}{2}x^{-1/2}dx$, we have $$\int_0^\infty 2e^{-(t^2-a)^2}dt .$$ ...
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### how to find the table integral that contains factorial of n?

I read in this page that" There are lots of definite integrals that depend on a parameter n∈N and whose result contains factorials of n " I am really struggle with this integral \begin{align*...
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### Calculate the integral by Riemann $\int_{0}^{1}\sqrt{x}\,dx$

Calculate the integral by Riemann $\displaystyle \int _{0}^{1}\sqrt{x} \, dx$ $$\displaystyle a_{k} =0+k\cdot \frac{1}{n} =\frac{k}{n}$$ $$\displaystyle \Delta x=\frac{1-0}{n} =\frac{1}{n}$$ \begin{...
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### Integrating $\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$

Let $\alpha>1$, $n \in \mathbb{N}$ and $q\geq0$. Which methods is possible to use to solve this integral? $$\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$$ I tried using the computer for especific ...
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### Taking an Integral with respect to time to find the work done on a rocket.

I was given a rocket problem for my Calculus 2 class and want to know if it's possible to take an integral of time to find the correct solution. I think it will be easier to explain if I show the ...
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### Understanding why even though derivative exists but the derivative is not integrable in the given x range

Why is the derivative which exists everywhere for $f(x) = x^2 \sin(\frac{1}{x^2})$ for $x \neq 0$ and $f(x) = 0$ for $x=0$ is not integrable over a region $[a,b ] \in \mathbb{R}$ having $0$ included ...
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