compagnia04

2022-01-05

Solve the integral:

$\int}_{0}^{2}\frac{dx}{\sqrt{x}(x-1)$

encolatgehu

Beginner2022-01-06Added 27 answers

You can break up the integral at any point or points you like. In this case, you could break it up into three integrals: pick a point c strictly between 0 and 1, and consider:

$\int}_{0}^{2}\frac{dx}{\sqrt{x}(x-1)}={\int}_{0}^{c}\frac{dx}{\sqrt{x}(x-1)}+{\int}_{c}^{1}\frac{dx}{\sqrt{x}(x-1)}+{\int}_{1}^{2}\frac{dx}{\sqrt{x}(x-1)$

$=\underset{a\to {0}^{+}}{lim}{\int}_{a}^{c}\frac{dx}{\sqrt{x}(x-1)}+\underset{b\to {1}^{-}}{lim}{\int}_{c}^{b}\frac{dx}{\sqrt{x}(x-1)}+\underset{d}{\overset{2}{lim}}\frac{dx}{\sqrt{x}(x-1)}$

The original improper integral exists if and only if each of the three improper integrals exist.

The original improper integral exists if and only if each of the three improper integrals exist.

Jenny Bolton

Beginner2022-01-07Added 32 answers

Consider the change of variables $t=\sqrt{x}$ . It transforms the interval $[0,2]$ into $[0,\sqrt{2}]$ and removes one of the problem zeroes. It is easy to see from the resulting expression that the integral diverges.

karton

Expert2022-01-11Added 613 answers

Try to solve the integral. You'll find whether it converges. For example,

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?