حل ∫ sqrt{2/x wrt x

$$\int{ \sqrt{ \frac{ 2 }{ x } } }d x$$

تقدير القيمة

$$2\sqrt{2x}+С$$

خطوات الحل

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$\sqrt{2}\int \sqrt{\frac{1}{x}}\mathrm{d}x$$

إعادة كتابة \frac{1}{\sqrt{x}} ك x^{-\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{-\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{1}{2}}}{\frac{1}{2}}. Simplify and convert from exponential to radical form.

$$\sqrt{2}\times 2\sqrt{x}$$

تبسيط.

$$2\sqrt{2}\sqrt{x}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$2\sqrt{2}\sqrt{x}+С$$

تفاضل w.r.t. x

$$\sqrt{\frac{2}{x}}$$