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Integration

Multiple, Sum and Difference Rules

The following rules also follow from the counterparts of differentiation:

  1. Constant multiple rule

    the indefinite integral of m f(x) dx = mthe indefinite integral of f(x) dx + c

  2. Sum rule

    the indefinite integral of (f(x) + g(x)) dx = the indefinite integral off(x) dx + the indefinite integral ofg(x) dx + c

  3. Difference rule

    the indefinite integral of (f(x) − g(x)) dx = the indefinite integral off(x) dxthe indefinite integral ofg(x) dx + c

Note that there are no general integration rules for products and quotients. There are rules for evaluating some special cases for them.

Examples

the indefinite integral of 6cos θ dθ = 6sin θ + c

the indefinite integral of15x4 dx = 15(1/5)x5 + c = 3x5 + c

the indefinite integral of(y + 2)dy = ½y2 + 2y + c

the indefinite integral of(e2+y − 8y) dy = e2the indefinite integral ofeydythe indefinite integral of8y dy = e2+y − 4y2 + c

the indefinite integral of 2 over x to the power 5 is negative 1 over (2 times x to the power 4)

Exercise

Do plenty of exercises until you feel confident of these rules with basic integrals.

Find the indefinite integrals:

indefinite integral =

<< Basic Integrals | Integration Index | Linear Substitution >>

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