Solve ∫ (x^2+4x-4/sqrt{x})}dθ

$$\int{ ( \frac{ { x }^{ 2 } +4x-4 }{ \sqrt{ x } } ) }d \theta$$

Evaluate

$$\frac{\theta \left(x^{2}+4x-4\right)}{\sqrt{x}}+С$$

Solution Steps

Find the integral of \frac{x^{2}+4x-4}{\sqrt{x}} using the table of common integrals rule \int a\mathrm{d}\theta =a\theta .

$$\frac{x^{2}+4x-4}{\sqrt{x}}\theta$$

Simplify.

$$\frac{\left(x^{2}+4x-4\right)\theta }{\sqrt{x}}$$

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

$$\frac{\left(x^{2}+4x-4\right)\theta }{\sqrt{x}}+С$$

Differentiate w.r.t. x

$$\frac{\theta \left(3x^{2}+4x+4\right)}{2x^{\frac{3}{2}}}$$