Integrals Resulting in Inverse Trigonometric Functions

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Evaluate the integral

Failed to parse (syntax error): {\displaystyle \[ \int\dfrac{dx}{\sqrt{4 - 9x^2}}.\]}

Solution

Substitute Failed to parse (syntax error): {\displaystyle \( u=3x\)} . Then Failed to parse (syntax error): {\displaystyle \( du=3\,dx\)} and we have

Failed to parse (syntax error): {\displaystyle \[ \int;\dfrac{dx}{\sqrt{4 - 9x^2}}=\dfrac{1}{3}\int\dfrac{du}{\sqrt{4 - u^2}}.\nonumber\]}

Applying the formula with Failed to parse (syntax error): {\displaystyle \( a=2,\)} we obtain

Failed to parse (syntax error): {\displaystyle \[ \int;\dfrac{dx}{\sqrt{4 - 9x^2}}=\dfrac{1}{3}\int\dfrac{du}{\sqrt{4 - u^2}}=\dfrac{1}{3}\arcsin \left(\dfrac{u}{2}\right)+C=\dfrac{1}{3}\arcsin \left(\dfrac{3x}{2}\right)+C.\]}

Resources

Integration into Inverse trigonometric functions using Substitution by The Organic Chemistry Tutor

Integrating using Inverse Trigonometric Functions by patrickJMT