Solve ∫ (2sqrt{x}-2/sqrt{2}) wrt x

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

Evaluate

$$\frac{4x^{\frac{3}{2}}}{3}-\sqrt{2}x+С$$

Solution Steps

Rationalize the denominator of \frac{2}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\int 2\sqrt{x}-\frac{2\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\mathrm{d}x$$

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

$$\int 2\sqrt{x}-\sqrt{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2\sqrt{x}\mathrm{d}x+\int -\sqrt{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int \sqrt{x}\mathrm{d}x-\int \sqrt{2}\mathrm{d}x$$

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}.

$$\frac{4x^{\frac{3}{2}}}{3}-\int \sqrt{2}\mathrm{d}x$$

Find the integral of \sqrt{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{4x^{\frac{3}{2}}}{3}-\sqrt{2}x$$

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.

$$\frac{4x^{\frac{3}{2}}}{3}-\sqrt{2}x+С$$

Differentiate w.r.t. x

$$\sqrt{2}\left(\sqrt{2x}-1\right)$$