Question

Find the integral. $$ \int \frac{5}{\sqrt{9-x^{2}}} d x $$

Short answer

The integral of the given function is \(5 \frac{x}{3} + C\).

Step-by-step solution

Trigonometric Substitution

Make the substitution \(x = 3 \sin{\theta}\) to simplify the expression under the square root. Calculate the differential \(dx = 3 \cos{\theta} d\theta\).

Expression Simplification

Substitute the value of \(x\) and \(dx\) into the integral, so it becomes \(\int \frac{5} {\sqrt{9-9\sin^{2}{\theta}}} * 3\cos{\theta} d\theta\). Simplify this expression to get \(\int 5\cos{\theta} d\theta\).

Integration

Integrate the expression \(\int5\cos{\theta} d\theta\) to get the integral in terms of \(\theta\). The integral of \(\cos{\theta}\) is \(\sin{\theta}\), so the result is \(5\sin{\theta} + C\), where \(C\) is the integration constant.

Back Substitution

Replace \(\theta\) with \(x\) using the relation \(x = 3 \sin{\theta}\), we get \(\theta = \arcsin(\frac{x}{3})\). Therefore, the final answer is \(5 \sin(\arcsin(\frac{x}{3})) + C = 5 \frac{x}{3} + C\).