Integration of other expressions

If, instead of having an expression with all terms in the form , like:

we instead have something like this:

…then we need to convert it into the form of the first expression.

Integrate

• Rewrite each term in the form :
  • stays the same - already in the right form.
  • can be rewritten as .
  • can be rewritten as .
• So the expression becomes:
• Now integrate this expression term-by-term:
• Adding them all together, we have:

Find

• Rewrite each term in the form :
  • can be rewritten as .
  • can be rewritten as .
  • can be rewritten as .
• So the expression becomes:
• Now integrate this expression term-by-term:
• Adding them all together, we have:

Find given that and

• Rewrite each term in the form :
  • can be rewritten as .
  • can be rewritten as .
  • stays the same - already in the right form.
• So the expression becomes:
• Now integrate this expression term-by-term:
• Adding them all together, we have:
• Now use the initial condition to find :
  • Set this equal to 7:
• Put this value of back into the expression for :

Find

• Rewrite each term in the form :
  • can be rewritten as .
  • can be rewritten as .
  • can be rewritten as .
• So the expression becomes:
• Now integrate this expression term-by-term:
• Adding them all together, we have: