Solve ∫ (sqrt{10}-sqrt{5})(sqrt{10}+sqrt{5})

$$\int ( \sqrt { 10 } - \sqrt { 5 } ) ( \sqrt { 10 } + \sqrt { 5 } )$$

Evaluate

$$5x+С$$

Solution Steps

Consider \left(\sqrt{10}-\sqrt{5}\right)\left(\sqrt{10}+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\int \left(\sqrt{10}\right)^{2}-\left(\sqrt{5}\right)^{2}\mathrm{d}x$$

The square of \sqrt{10} is 10.

$$\int 10-\left(\sqrt{5}\right)^{2}\mathrm{d}x$$

The square of \sqrt{5} is 5.

$$\int 10-5\mathrm{d}x$$

Subtract 5 from 10 to get 5.

$$\int 5\mathrm{d}x$$

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$5x$$

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.

$$5x+С$$

Differentiate w.r.t. x

$$5$$