Harder Substitution (Cambridge International A-Level Mathematics)

Harder substitution

What is integration by substitution?

• Make sure you are familiar with Reverse Chain Rule and Substitution (RCR)
• In more difficult questions the substitution will be given to you

How do I integrate using a given substitution?

• The process is very much the same as before but expect the algebra to be trickier

• STEP 1: Differentiate the substitution and rearrange

  • $\frac{du}{dx}$ here can be treated like a fraction (eg × dx to get rid of fractions)

• STEP 2: Replace all parts of the integral

  • all x terms should be replaced with equivalent u terms, including dx
  • if a definite integral change the limits from x to u too

• STEP 3: Integrate and either

  • substitute x back in or
  • evaluate the definite integral using the u limits

• STEP 4: Find c, the constant of integration, if needed

Worked Example