U-Substitution for Integration

Overview

• U-substitution is a method for finding antiderivatives, especially useful for definite integrals.
• The key steps involve identifying the substitution variable ( u ) and its derivative ( du ), and then substituting back after integration.

Example Problems

Problem 1: ( \int 4x(x^2 + 5)^3 , dx )

1. Set ( u = x^2 + 5 ):
   • ( du = 2x , dx )
   • Solving for ( dx ): ( dx = \frac{du}{2x} )
2. Substitute:
   • ( 4x(x^2 + 5)^3 , dx = 2 \int u^3 , du )
3. Integrate:
   • ( 2 \times \frac{u^4}{4} + C = \frac{1}{2}u^4 + C )
4. Back Substitute:
   • ( \frac{1}{2}(x^2 + 5)^4 + C )

Problem 2: ( \int 8 \cos(4x) , dx )

1. Set ( u = 4x ):
   • ( du = 4 , dx )
   • ( dx = \frac{du}{4} )
2. Substitute:
   • ( 8 \int \cos(u) \cdot \frac{du}{4} = 2 \int \cos(u) , du )
3. Integrate:
   • ( 2 \sin(u) + C )
4. Back Substitute:
   • ( 2 \sin(4x) + C )

Problem 3: ( \int x^3 e^{x^4} , dx )

1. Set ( u = x^4 ):
   • ( du = 4x^3 , dx )
   • ( dx = \frac{du}{4x^3} )
2. Substitute and Simplify:
   • ( \int e^u \cdot \frac{du}{4} = \frac{1}{4} \int e^u , du )
3. Integrate:
   • ( \frac{1}{4} e^u + C )
4. Back Substitute:
   • ( \frac{1}{4} e^{x^4} + C )

Problem 4: ( \int 8x \sqrt{40 - 2x^2} , dx )

1. Set ( u = 40 - 2x^2 ):
   • ( du = -4x , dx )
   • ( dx = \frac{du}{-4x} )
2. Substitute and Simplify:
   • ( -2 \int u^{1/2} , du )
3. Integrate using Power Rule:
   • ( -\frac{4}{3}u^{3/2} + C )
4. Back Substitute:
   • ( -\frac{4}{3}(40 - 2x^2)^{3/2} + C )

Additional Examples

Problem 5: ( \int \frac{x^3}{(2 + x^4)^2} , dx )

1. Set ( u = 2 + x^4 ):
   • ( du = 4x^3 , dx )
   • ( dx = \frac{du}{4x^3} )
2. Substitute and Cancel:
   • ( \frac{1}{4} \int u^{-2} , du )
3. Integrate:
   • ( -\frac{1}{4u} + C )
4. Back Substitute:
   • ( -\frac{1}{4(2 + x^4)} + C )

Problem 6: ( \int \sin^4(x) \cos(x) , dx )

1. Set ( u = \sin(x) ):
   • ( du = \cos(x) , dx )
2. Substitute:
   • ( \int u^4 , du )
3. Integrate:
   • ( \frac{1}{5}u^5 + C )
4. Back Substitute:
   • ( \frac{1}{5}\sin^5(x) + C )

Key Tips for U-Substitution

• Identify ( u ) such that its derivative ( du ) matches terms in the integrand.
• Always solve for ( dx ) in terms of ( du ).
• Substitute back all instances of ( u ) at the end.