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Exercise

$\frac{u}{\sqrt{1+u^2}\left(1-\sqrt{1+u^2}\right)}=\frac{dx}{du}\cdot\frac{1}{x}$

Step-by-step Solution

1

Group the terms of the differential equation. Move the terms of the $u$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

$\frac{u}{\sqrt{1+u^2}\left(1-\sqrt{1+u^2}\right)}du=\frac{1}{x}dx$
2

Integrate both sides of the differential equation, the left side with respect to $u$, and the right side with respect to $x$

$\int\frac{u}{\sqrt{1+u^2}\left(1-\sqrt{1+u^2}\right)}du=\int\frac{1}{x}dx$
3

Solve the integral $\int\frac{u}{\sqrt{1+u^2}\left(1-\sqrt{1+u^2}\right)}du$ and replace the result in the differential equation

$-\ln\left|1-\sqrt{1+u^2}\right|=\int\frac{1}{x}dx$
4

Multiply both sides of the equation by $-1$

$\ln\left|1-\sqrt{1+u^2}\right|=-\int\frac{1}{x}dx$
5

Solve the integral $-\int\frac{1}{x}dx$ and replace the result in the differential equation

$\ln\left|1-\sqrt{1+u^2}\right|=-\ln\left|x\right|+C_0$

Final answer to the exercise

$\ln\left|1-\sqrt{1+u^2}\right|=-\ln\left|x\right|+C_0$

Try other ways to solve this exercise

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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