Integral of $$$- \frac{10}{\sqrt{49 - 36 x^{2}}}$$$

Find $$$\int \left(- \frac{10}{\sqrt{49 - 36 x^{2}}}\right)\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-10$$$ and $$$f{\left(x \right)} = \frac{1}{\sqrt{49 - 36 x^{2}}}$$$:

$${\color{red}{\int{\left(- \frac{10}{\sqrt{49 - 36 x^{2}}}\right)d x}}} = {\color{red}{\left(- 10 \int{\frac{1}{\sqrt{49 - 36 x^{2}}} d x}\right)}}$$

Let $$$x=\frac{7 \sin{\left(u \right)}}{6}$$$.

Then $$$dx=\left(\frac{7 \sin{\left(u \right)}}{6}\right)^{\prime }du = \frac{7 \cos{\left(u \right)}}{6} du$$$ (steps can be seen »).

Also, it follows that $$$u=\operatorname{asin}{\left(\frac{6 x}{7} \right)}$$$.

Thus,

$$$\frac{1}{\sqrt{49 - 36 x^{2}}} = \frac{1}{\sqrt{49 - 49 \sin^{2}{\left( u \right)}}}$$$

Use the identity $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

$$$\frac{1}{\sqrt{49 - 49 \sin^{2}{\left( u \right)}}}=\frac{1}{7 \sqrt{1 - \sin^{2}{\left( u \right)}}}=\frac{1}{7 \sqrt{\cos^{2}{\left( u \right)}}}$$$

Assuming that $$$\cos{\left( u \right)} \ge 0$$$, we obtain the following:

$$$\frac{1}{7 \sqrt{\cos^{2}{\left( u \right)}}} = \frac{1}{7 \cos{\left( u \right)}}$$$

Integral becomes

$$- 10 {\color{red}{\int{\frac{1}{\sqrt{49 - 36 x^{2}}} d x}}} = - 10 {\color{red}{\int{\frac{1}{6} d u}}}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=\frac{1}{6}$$$:

$$- 10 {\color{red}{\int{\frac{1}{6} d u}}} = - 10 {\color{red}{\left(\frac{u}{6}\right)}}$$

Recall that $$$u=\operatorname{asin}{\left(\frac{6 x}{7} \right)}$$$:

$$- \frac{5 {\color{red}{u}}}{3} = - \frac{5 {\color{red}{\operatorname{asin}{\left(\frac{6 x}{7} \right)}}}}{3}$$

Therefore,

$$\int{\left(- \frac{10}{\sqrt{49 - 36 x^{2}}}\right)d x} = - \frac{5 \operatorname{asin}{\left(\frac{6 x}{7} \right)}}{3}$$

Add the constant of integration:

$$\int{\left(- \frac{10}{\sqrt{49 - 36 x^{2}}}\right)d x} = - \frac{5 \operatorname{asin}{\left(\frac{6 x}{7} \right)}}{3}+C$$

$$$\int \left(- \frac{10}{\sqrt{49 - 36 x^{2}}}\right)\, dx = - \frac{5 \operatorname{asin}{\left(\frac{6 x}{7} \right)}}{3} + C$$$A