Question

Evaluate the following integrals: $$ \int \frac{\mathrm{dx}}{\sqrt{2 \mathrm{ax}-\mathrm{x}^{2}}} $$

Short answer

Question: Evaluate the integral: $$ \int \frac{\mathrm{dx}}{\sqrt{2 \mathrm{ax}-\mathrm{x}^{2}}} $$ Answer: $$ \frac{x}{a}+C $$

Step-by-step solution

Determine the trigonometric substitution

Since the expression \(2ax-x^2\) is inside the square root function, we will use the sine trigonometric function. The substitution will be: $$ x = a\sin(\theta) $$

Calculate the differential dx

We need to find the differential to replace dx in the integral expression. Differentiate both sides of the expression with respect to \(\theta\): $$ \frac{\mathrm{dx}}{\mathrm{d}\theta} = a\cos(\theta) $$ Now, multiply both sides by \(\mathrm{d}\theta\) to find \(\mathrm{dx}\): $$ \mathrm{dx} = a\cos(\theta) \, \mathrm{d}\theta $$

Replace x and dx in the integral expression

Replace x with the sin substitution and dx with the cos substitution: $$ \int \frac{\mathrm{dx}}{\sqrt{2a(a\sin(\theta))-a^2\sin^2(\theta)}} = \int\frac{a\cos(\theta) \, \mathrm{d}\theta}{\sqrt{2a^2\sin(\theta)-a^2\sin^2(\theta)}} $$

Simplify the expression and find the integral

Simplify the expression in the integral: $$ \int\frac{a\cos(\theta) \, \mathrm{d}\theta}{\sqrt{a^2(2\sin(\theta)-\sin^2(\theta))}} = \int\frac{a\cos(\theta) \, \mathrm{d}\theta}{a\sqrt{2\sin(\theta)-\sin^2(\theta)}} $$ Now, we can cancel out the factor a in the numerator and denominator: $$ \int\frac{\cos(\theta) \, \mathrm{d}\theta}{\sqrt{2\sin(\theta)-\sin^2(\theta)}} $$

Evaluate the integral

Now we have a more manageable integral to solve: $$ \int\frac{\cos(\theta) \, \mathrm{d}\theta}{\sqrt{2\sin(\theta)-\sin^2(\theta)}} = \int\frac{\cos(\theta) \, \mathrm{d}\theta}{\sqrt{\sin(\theta)(2-\sin(\theta))}} $$ Imultiply and divide by \(\sqrt{2-\sin(\theta)}\) $$ = \int\frac{\cos(\theta)\sqrt{2-\sin(\theta)} \, \mathrm{d}\theta}{\sqrt{\sin(\theta)(2-\sin(\theta))\cdot(2-\sin(\theta))}} $$ Now the denominator simplifies to \(\sqrt{2\sin(\theta)-\sin^2(\theta)}\) $$ = \int\frac{\cos(\theta)\sqrt{2-\sin(\theta)} \, \mathrm{d}\theta}{\sqrt{2\sin(\theta)-\sin^2(\theta)}} $$ Now apply a new substitution: \(u=\sin(\theta)\Rightarrow \mathrm{d}u = \cos(\theta) \, \mathrm{d}\theta\) $$ = \int\frac{\mathrm{d}u\sqrt{2-u}}{\sqrt{u(2-u)}} $$ The integral simplifies to: $$ = \int\mathrm{d}u $$ After integrating, we have: $$ u+C=\sin(\theta)+C $$ Now, substitute back the original variable, x: $$ \sin(\theta)=\frac{x}{a} \Rightarrow \int \frac{\mathrm{dx}}{\sqrt{2 \mathrm{ax}-\mathrm{x}^{2}}} = \frac{x}{a}+C $$