Evaluating Definite Integrals Using U-Substitution

Introduction

• Focus on evaluating definite integrals using the technique of u-substitution.
• Start with an example to find the value of the definite integral of (2x(x^2 + 4)^2) from 0 to 2.

Example 1

Steps

1. U-Substitution:

   • Set (u = x^2 + 4)
   • Calculate (du = 2x , dx)
   • Isolate (dx) by dividing both sides by (2x): (dx = \frac{du}{2x})

2. Change of Variables:

   • Substitute (x^2 + 4) with (u) and (dx) with (\frac{du}{2x}).
   • Expression becomes (u^2 \frac{du}{2x})

3. Adjust Limits:

   • Lower limit: (x = 0), (u = 0^2 + 4 = 4)
   • Upper limit: (x = 2), (u = 2^2 + 4 = 8)

4. Integration:

   • Integrate (u^2) from 4 to 8.
   • Antiderivative: (\frac{u^3}{3})
   • Evaluate as: (\frac{8^3}{3} - \frac{4^3}{3})
   • Calculations:
     • (8^3 = 512)
     • (4^3 = 64)
     • Result: (\frac{448}{3})

Key Concept

• When changing variables from (x) to (u), also change the limits accordingly.

Example 2

• Evaluate (4x \sqrt{16 - x^2}) from 0 to 4 using u-substitution.

Steps

1. U-Substitution:

   • Set (u = 16 - x^2)
   • Calculate (du = -2x , dx)
   • Isolate (dx): (dx = \frac{du}{-2x})

2. Replace Variables:

   • Expression becomes (4x \sqrt{u} \cdot \frac{du}{-2x})
   • Simplifies to (-2 \sqrt{u} , du)

3. Adjust Limits:

   • Lower limit: (x = 0), (u = 16)
   • Upper limit: (x = 4), (u = 0)

4. Integration:

   • Integrate (-2u^{1/2}) from 16 to 0.
   • Antiderivative: (-\frac{4}{3} u^{3/2})
   • Evaluate:
     • (16^{3/2} = 64)
     • Result: (\frac{256}{3})

Example 3

• Evaluate (2x (1 + x^2)^3) from 1 to 2 using u-substitution.

Steps

1. U-Substitution:

   • Set (u = 1 + x^2)
   • (du = 2x , dx)

2. Replace Variables:

   • Expression becomes (\frac{1}{u^3}) from 2 to 5.

3. Adjust Limits:

   • Lower limit: (x = 1), (u = 2)
   • Upper limit: (x = 2), (u = 5)

4. Integration:

   • Antiderivative: (-\frac{1}{2} u^{-2})
   • Evaluate:
     • Result: (\frac{21}{200})

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Conclusion

• U-substitution allows simplification and solving of definite integrals by changing variables and adjusting the limits of integration.
• Important to replace all x-terms with u-terms and adjust the integration limits.
• Practice with various expressions to get comfortable with the technique.