Solve ∫ sqrt{x}(sqrt{lambda}+3) wrt x

$$\int \sqrt { x } ( \sqrt { \lambda } + 3 ) d x$$

Evaluate

$$\frac{2\left(\sqrt{\lambda }+3\right)x^{\frac{3}{2}}}{3}+С$$

Solution Steps

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

$$\left(\sqrt{\lambda }+3\right)\int \sqrt{x}\mathrm{d}x$$

Rewrite \sqrt{x} as 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{3}{2}}}{\frac{3}{2}}. Simplify.

$$\left(\sqrt{\lambda }+3\right)\times \frac{2x^{\frac{3}{2}}}{3}$$

Simplify.

$$\frac{2\left(\sqrt{\lambda }+3\right)x^{\frac{3}{2}}}{3}$$

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.

$$\frac{2\left(\sqrt{\lambda }+3\right)x^{\frac{3}{2}}}{3}+С$$

Differentiate w.r.t. x

$$\sqrt{x}\left(\sqrt{\lambda }+3\right)$$