Solve ∫ (2sqrt{x}+3/x) wrt x

$$\int ( 2 \sqrt { x } + \frac { 3 } { x } ) d x$$

Evaluate

$$3\ln(x)+\frac{4x^{\frac{3}{2}}}{3}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2\sqrt{x}\mathrm{d}x+\int \frac{3}{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int \sqrt{x}\mathrm{d}x+3\int \frac{1}{x}\mathrm{d}x$$

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. Multiply 2 times \frac{2x^{\frac{3}{2}}}{3}.

$$\frac{4x^{\frac{3}{2}}}{3}+3\int \frac{1}{x}\mathrm{d}x$$

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$\frac{4x^{\frac{3}{2}}}{3}+3\ln(|x|)$$

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.

$$\frac{4x^{\frac{3}{2}}}{3}+3\ln(x)+С$$

Differentiate w.r.t. x

$$2\sqrt{x}+\frac{3}{x}$$