Lecture Notes on Integral Calculus

Overview

• Discussing an integral from the book by DailyMath.
• The integral is represented as:
  [ f(\alpha) = \int_0^{\infty} e^{-x^2} \cos(\alpha x) , dx ]
• The goal is to evaluate this integral for different values of ( \alpha ).

Differentiation Under the Integral Sign

• Differentiate both sides with respect to ( \alpha ):
  [ f'(\alpha) \text{ and } \frac{d}{d\alpha} \int_0^{\infty} e^{-x^2} \cos(\alpha x) , dx ]

Applying the Leibniz Rule

• Use the Leibniz rule to differentiate under the integral:
  [ \frac{d}{d\alpha} \cos(\alpha x) = -x \sin(\alpha x) ]
• Resulting in:
  [ f'(\alpha) = \int_0^{\infty} e^{-x^2} (-x \sin(\alpha x)) , dx ]
• Simplifying:
  [ f'( abla) = -\int_0^{\infty} x e^{-x^2} \sin(\alpha x) , dx ]

Integration by Parts Setup

• Set up for integration by parts:
  • Let:
    • ( u = \sin(\alpha x) )
    • ( dv = e^{-x^2} dx )
• Compute derivatives:
  • ( du = \alpha \cos(\alpha x) , dx )
  • ( v = -\frac{1}{2} e^{-x^2} )

Integration Steps

• Apply integration by parts:
  • First term evaluates to zero at limits (as ( e^{-x^2} ) approaches zero).
  • Second term results in: [ -\frac{\alpha}{2} \int_0^{\infty} e^{-x^2} \sin(\alpha x) , dx ]

Resulting Differential Equation

• The equation simplifies to: [ f'( abla) = -\frac{\alpha}{2} f(\alpha) ]
• This is a separable differential equation: [ \frac{df}{f} = -\frac{\alpha}{2} d\alpha ]

Integrating Both Sides

• Integrating gives: [ \ln |f| = -\frac{\alpha^2}{4} + C ]
  • Where C is a constant.
• Exponentiating results in: [ f(\alpha) = C_2 e^{-\frac{\alpha^2}{4}} ]

Finding the Constant ( C_2 )

• Substitute ( \alpha = 0 ) in the original integral:
  [ f(0) = \int_0^{\infty} e^{-x^2} , dx = \frac{\sqrt{\pi}}{2} ]
• Hence ( C_2 = \frac{\sqrt{\pi}}{2} )

Final Result

• Therefore, the integral evaluates to: [ f(\alpha) = \frac{\sqrt{\pi}}{2} e^{-\frac{\alpha^2}{4}} ]
• For ( \alpha = 5 ):
  [ f(5) = \frac{\sqrt{\pi}}{2} e^{-\frac{25}{4}} ]

Conclusion

• The integral can be computed for any value of ( \alpha ) using the derived formula.