Solve ∫ 15/21sqrt{14^2-7^2}

$$\int \frac { 15 } { 21 \sqrt { 14 ^ { 2 } - 7 ^ { 2 } } }$$

Evaluate

$$\frac{5\sqrt{3}x}{147}+С$$

Solution Steps

Calculate 14 to the power of 2 and get 196.

$$\int \frac{15}{21\sqrt{196-7^{2}}}\mathrm{d}x$$

Calculate 7 to the power of 2 and get 49.

$$\int \frac{15}{21\sqrt{196-49}}\mathrm{d}x$$

Subtract 49 from 196 to get 147.

$$\int \frac{15}{21\sqrt{147}}\mathrm{d}x$$

Factor 147=7^{2}\times 3. Rewrite the square root of the product \sqrt{7^{2}\times 3} as the product of square roots \sqrt{7^{2}}\sqrt{3}. Take the square root of 7^{2}.

$$\int \frac{15}{21\times 7\sqrt{3}}\mathrm{d}x$$

Multiply 21 and 7 to get 147.

$$\int \frac{15}{147\sqrt{3}}\mathrm{d}x$$

Rationalize the denominator of \frac{15}{147\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\int \frac{15\sqrt{3}}{147\left(\sqrt{3}\right)^{2}}\mathrm{d}x$$

The square of \sqrt{3} is 3.

$$\int \frac{15\sqrt{3}}{147\times 3}\mathrm{d}x$$

Cancel out 3 in both numerator and denominator.

$$\int \frac{5\sqrt{3}}{3\times 49}\mathrm{d}x$$

Multiply 3 and 49 to get 147.

$$\int \frac{5\sqrt{3}}{147}\mathrm{d}x$$

Find the integral of \frac{5\sqrt{3}}{147} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{5\sqrt{3}}{147}x$$

Simplify.

$$\frac{5\sqrt{3}x}{147}$$

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.

$$\frac{5\sqrt{3}x}{147}+С$$

Differentiate w.r.t. x

$$\frac{5 \sqrt{3}}{147} = 0.0589132927744516$$