3.6.3 Integration

Again, only basic integration is required for the number sense test. The technique for integrating is essentially taking the derivative backwards (or anti-derivative) and then plugging in the limits of integration. The following shows a generic polynomial being integrated:

$$\int_{a}^{b} a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x^1 + a_0 x^0 dx = F(x) = \left[ \frac{a_n}{n+1} x^{n+1} + \frac{a_{n-1}}{n} x^n + \dots + \frac{a_1}{2} x^2 + \frac{a_0}{1} x^1 \right]_{a}^{b} = F(b) - F(a)$$

Let’s look at an example:

Problem: Evaluate $\int_{0}^{2} 3x^2 - x dx$. Solution: $\int_{0}^{2} 3x^2 - x dx = \left[ x^3 - \frac{1}{2}x^2 \right]_{0}^{2} = (2^3 - \frac{1}{2}2^2) - (0^3 - \frac{1}{2} \cdot 0) = 6$

Again, you can apply the table in the previous section for computing integrals of functions (just go in reverse).

To end this section on Integration, there is one special case when integrating, such that the integral is trivial, and that is:

$$\int_{-a}^{a} \text{Odd Function } dx = 0$$

So when you are integrating an odd function who’s limits are negatives of each other, the result is zero. Let’s look at an example of where to apply this:

$$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sin(x) dx = 0$$

Since sine is an odd function, the integral (with the appropriate negative limits) is simply zero!

The following are some more practice problems concerning integration:

Problem Set 3.6.3