Solve ∫ frac{-sqrt{2}}{-2}frac{sqrt{2^224^2-24^2}x^2}{{x}^2 wrt x

$$\int{ \frac{ - \sqrt{ 2 } }{ -2 } \frac{ \sqrt{ { 2 }^{ 2 } { 24 }^{ 2 } - { 24 }^{ 2 } } { x }^{ 2 } }{ { x }^{ 2 } } }d x$$

Evaluate

$$12\sqrt{6}x+С$$

Solution Steps

Cancel out x^{2} in both numerator and denominator.

$$\int \frac{-\sqrt{2}}{-2}\sqrt{2^{2}\times 24^{2}-24^{2}}\mathrm{d}x$$

Express \frac{-\sqrt{2}}{-2}\sqrt{2^{2}\times 24^{2}-24^{2}} as a single fraction.

$$\int \frac{\left(-\sqrt{2}\right)\sqrt{2^{2}\times 24^{2}-24^{2}}}{-2}\mathrm{d}x$$

Calculate 2 to the power of 2 and get 4.

$$\int \frac{\left(-\sqrt{2}\right)\sqrt{4\times 24^{2}-24^{2}}}{-2}\mathrm{d}x$$

Calculate 24 to the power of 2 and get 576.

$$\int \frac{\left(-\sqrt{2}\right)\sqrt{4\times 576-24^{2}}}{-2}\mathrm{d}x$$

Multiply 4 and 576 to get 2304.

$$\int \frac{\left(-\sqrt{2}\right)\sqrt{2304-24^{2}}}{-2}\mathrm{d}x$$

Calculate 24 to the power of 2 and get 576.

$$\int \frac{\left(-\sqrt{2}\right)\sqrt{2304-576}}{-2}\mathrm{d}x$$

Subtract 576 from 2304 to get 1728.

$$\int \frac{\left(-\sqrt{2}\right)\sqrt{1728}}{-2}\mathrm{d}x$$

Factor 1728=24^{2}\times 3. Rewrite the square root of the product \sqrt{24^{2}\times 3} as the product of square roots \sqrt{24^{2}}\sqrt{3}. Take the square root of 24^{2}.

$$\int \frac{\left(-\sqrt{2}\right)\times 24\sqrt{3}}{-2}\mathrm{d}x$$

Divide \left(-\sqrt{2}\right)\times 24\sqrt{3} by -2 to get \left(-\sqrt{2}\right)\left(-12\right)\sqrt{3}.

$$\int \left(-\sqrt{2}\right)\left(-12\right)\sqrt{3}\mathrm{d}x$$

Multiply -1 and -12 to get 12.

$$\int 12\sqrt{2}\sqrt{3}\mathrm{d}x$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\int 12\sqrt{6}\mathrm{d}x$$

Find the integral of 12\sqrt{6} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$12\sqrt{6}x$$

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.

$$12\sqrt{6}x+С$$

Differentiate w.r.t. x

$$12 \sqrt{6} = 29.393876913$$