Rules of Integration

Question 1

When is the ** Reverse Power Rule** used?

Answer 1

The Reverse Power Rule can be used anytime there is a basic polynomial being integrated. Examples include:

∫ 14x³ - 2x² + 5 dx
∫ eˣ + 7x² + √x dx

Question 2

When is the Completing the Square Method of integration used?

Answer 2

When there’s a quadratic polynomial in the denominator and no variables in the numerator:

Ex: ∫ 1/(3x² + 6x +78) dx = (1/15)arctan((x + 1)/5)

∫ 1/√(-x² + 10x +11) dx = arcsin((x - 5)/6)

Question 3

When is the *Long Division Method** of Integration used?

Answer 3

When the function is a rational expression and the degree in the numerator is greater than or equal to the degree in the denominator.

Examples:

∫ (x - 5)/(-2x + 2) dx = ∫ ( (-1/2) + 2/(x - 1) ) dx

Question 4

Explain and define the Integration rule for:

∫ 1/(x + 2) dx

Answer 4

Since the definite integral of 1/x is ln|x| we will do u substitution.

u = x + 2
du = 1dx

substituting u we get → ∫ 1/u du and after integrating we get:

ln|u| which becomes ln|x + 2|

Question 5

Explain and define the Integration rule for:

∫ 1/(5x + 2) dx

Answer 5

Since the definite integral of 1/x is ln|x| we will do u substitution.

u = 5x + 2
du = 5dx

To add a 5 multiple into the integral without offsetting the function we will need to place a 1/5 outside of the integral:

∫ 1/(5x + 2) dx → 1/5 ∫ 1/(5x + 2) 5dx

Substituting u we get → 1/5 ∫ 1/(u) du and after integrating we get:

ln|u|/5 which becomes ln|5x + 2|/5