Example 1: Find ∫ x^3 dx.
Using the Power Rule, we get
∫^ x
(^3) dx = + C
Don’t forget the constant C, or your teachers will take points off for that too!
Example 2: Find ∫ x−3 dx.
Think of a constant as something that is always there, and you
won’t forget it.The Power Rule works with negative exponents too.
∫^ x
−3 dx = + CNot terribly hard, is it? Now it’s time for a few more rules that look remarkably similar to the rules for
derivatives that we saw in Chapter 6.
∫^ kf(x) dx = k^ ∫^ f(x) dx
∫ [f(x) + g(x)] dx = ∫^ f(x) dx + ∫^ g(x) dx
∫kdx = kx + C
Here are a few more examples to make you an expert.
Example 3: (^) ∫ 5 dx = 5x + C
Example 4: (^) ∫ 7 x^3 dx = + C
Example 5: (^) ∫ (3x^2 + 2x) dx = x^3 + x^2 + C