Integration Rules

Question 1

What is the formula to integrate by parts?

Answer 1

• (Integral) f(x) = UV - (integral) (v x du/dx)

Question 2

How do you integrate by inspection?

Answer 2

• You can only use it where there are no variables outside of the bracket.
• It is quite difficult to do, and is prone to error, so in most cases using a substitution would be a better idea.

Question 3

How do you integrate by substitution?

Answer 3

• Assign a variable, such as u, to the section of the expression which is proving difficult to integrate. Such as root x.
• Differentiate U on it’s own to get du/dx
• You then need to get all parts of the expression in terms of U, to do this rearrange the equation for U to make it the subject to eliminate any X variables still in the expression.
• Once the expression is in terms of U only, it can then be integrated normally.

Question 4

When do you use integration by parts?

Answer 4

• When you have two expressions multiplied together.
• E.g y =(4x²+5x)(sinx)

Question 5

What check should be performed before using integration by parts to help catch any sneaky bois?

Answer 5

• Check that the expressions cannot be easily expanded, to avoid using an overcomplicated method.

Question 6

In what circumstance do you have to replace the u values as x values after integrating by substitution?

Answer 6

• The integral must be converted back into terms of x when no limits are being applied.