Properties of Integrals

Notation

If and is integrable on , then:

Properties

Suppose and are integrable on .

Addition

will be integrable on .

Proof Hint

• Prove is integrable using:
  • )
  • )
• Start with and show
• Start with and show

Constant multiplication

Suppose . will be integrable .

Proof Hint

• Prove for . Use
• Prove for
• Using the above results, proof for is apparent

Bounds

If on :

If on :

Modulus

will be integrable on .

Proof Hint

Start with . And integrate both sides.

Multiple

will be integrable on .

Proof Hint

• Suppose is bounded by
• Prove is integrable (Use )
• is integrable because:

Max, Min

and are integrable.

Where and functions are defined as: