Math Magic: Integral Explorer

The Cosine Conundrum

\[ \int_{0}^{\pi/2} \frac{dx}{1 + 5 \cos^2 x} \]

Divide by cos²x

We transform the integral by dividing numerator and denominator by cos²x: \[ \frac{1}{1 + 5\cos^2x} = \frac{\sec^2x}{\sec^2x + 5} = \frac{\sec^2x}{6 + 5\tan^2x} \]

Substitution Magic

Let t = \tan x, then dt = \sec^2x dx.
When x = 0, t = 0
When x = \pi/2, t \to \infty
The integral transforms to: \[ \int_{0}^{\infty} \frac{dt}{6 + 5t^2} \]

Factor Out Constants

We factor out 5 from the denominator: \[ \frac{1}{5} \int_{0}^{\infty} \frac{dt}{\frac{6}{5} + t^2} \]

Standard Integral Form

We recognize the standard form: \[ \int \frac{dt}{a^2 + t^2} = \frac{1}{a} \tan^{-1}\left(\frac{t}{a}\right) + C \] Where a^2 = \frac{6}{5}, so a = \sqrt{\frac{6}{5}}

Evaluate the Integral

Applying the standard form: \[ \frac{1}{5} \times \frac{1}{\sqrt{6/5}} \left[ \tan^{-1}\left(\frac{t}{\sqrt{6/5}}\right) \right]_{0}^{\infty} \] \[ = \frac{1}{\sqrt{30}} \left( \frac{\pi}{2} - 0 \right) \]

The Magical Final Answer: \[ \frac{\pi}{2\sqrt{30}} \approx 0.2866 \]

The Sine Challenge

\[ \int_{0}^{\pi/2} \frac{dx}{5 + 4 \sin^2 x} \]

Divide by cos²x

We transform the integral by dividing numerator and denominator by cos²x: \[ \frac{1}{5 + 4\sin^2x} = \frac{\sec^2x}{5\sec^2x + 4\tan^2x} = \frac{\sec^2x}{5 + 9\tan^2x} \]

Substitution Magic

Let t = \tan x, then dt = \sec^2x dx.
When x = 0, t = 0
When x = \pi/2, t \to \infty
The integral transforms to: \[ \int_{0}^{\infty} \frac{dt}{5 + 9t^2} \]

Factor Out Constants

We factor out 9 from the denominator: \[ \frac{1}{9} \int_{0}^{\infty} \frac{dt}{\frac{5}{9} + t^2} \]

Standard Integral Form

We recognize the standard form: \[ \int \frac{dt}{a^2 + t^2} = \frac{1}{a} \tan^{-1}\left(\frac{t}{a}\right) + C \] Where a^2 = \frac{5}{9}, so a = \frac{\sqrt{5}}{3}

Evaluate the Integral

Applying the standard form: \[ \frac{1}{9} \times \frac{1}{\sqrt{5}/3} \left[ \tan^{-1}\left(\frac{t}{\sqrt{5}/3}\right) \right]_{0}^{\infty} \] \[ = \frac{1}{3\sqrt{5}} \left( \frac{\pi}{2} - 0 \right) \]

The Magical Final Answer: \[ \frac{\pi}{6\sqrt{5}} \approx 0.2342 \]