(3) Integration into Inverse trigonometric functions using Substitution

Integrating Functions Leading to Inverse Trigonometric Functions

Key Integration Formulas

• The following integration formulas for inverse trigonometric functions are essential:

  • Arcsin Formula:

    [ \int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin\left(\frac{u}{a}\right) + C ]

  • Arctan Formula:

    [ \int \frac{du}{a^2 + u^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C ]

  • Arcsinh Formula:

    [ \int \frac{du}{u\sqrt{u^2 + a^2}} = \frac{1}{a} \arcsinh\left(\frac{u}{a}\right) + C ]

Example Problems

1. Anti-derivative of ( \frac{dx}{\sqrt{16 - x^2}} )

• Identify parameters:

  • ( u = x )

  • ( du = dx )

  • ( a = 4 )

• Using the arcsin formula:

  [ \int \frac{dx}{\sqrt{16 - x^2}} = \arcsin\left(\frac{x}{4}\right) + C ]

2. Anti-derivative of ( \frac{3}{25 + x^2} dx )

• Identify parameters:

  • ( u = x )

  • ( a = 5 )

• Using the arctan formula:

  [ \int \frac{3}{25 + x^2} dx = \frac{3}{5} \arctan\left(\frac{x}{5}\right) + C ]

3. Anti-derivative of ( \frac{8}{x \sqrt{4x^2 - 1}} dx )

• Identify parameters:

  • ( u = 2x )

  • ( a = 1 )

• Using the arcsinh formula:

  [ 8 \int \frac{du}{u \sqrt{u^2 - 1}} = 8 \cdot 1 \cdot \arcsinh(2x) + C ]

• Final answer:

  [ 8 \arcsinh(2x) + C ]

4. Anti-derivative of ( \frac{x}{\sqrt{1 - x^4}} dx )

• Identify parameters:

  • ( u = x^2 )

  • ( a = 1 )

• The formula applies:

  [ \int \frac{x}{\sqrt{1 - x^4}} dx = \frac{1}{2} \arcsin(x^2) + C ]

5. Anti-derivative of ( \frac{x}{x^4 + 36} dx )

• Identify parameters:

  • ( u = x^2 )

  • ( a = 6 )

• The expression is associated with arctan:

  [ \int \frac{x}{x^4 + 36} dx = \frac{1}{12} \arctan\left(\frac{x^2}{6}\right) + C ]

6. Anti-derivative of ( \frac{dx}{x^2 - 4x + 7} )

• Complete the square:

  [ (x - 2)^2 + 3 ]

• Thus, using the arctan formula:

  [ \frac{1}{\sqrt{3}} \arctan\left(\frac{x - 2}{\sqrt{3}}\right) + C ]

7. Anti-derivative of ( \frac{dx}{x^4 + 2x^2 + 2} )

• Complete the square:

  [ (x^2 + 1)^2 + 1 ]

• Final expression gives:

  [ \frac{1}{2} \arctan(x^2 + 1) + C ]

General Tips

• For each integration problem:

  1. Identify the form (arcsin, arctan, or arcsinh).

  2. Make appropriate substitutions for (u) and (a).

  3. Always include the constant of integration (+ C) for indefinite integrals.