Let us look at some examples now.


Example 1.Find (x - 1) (x + 2) dx.

Expanding and integrating term by term, we obtain

(x - 1) (x + 2) dx = (x² + x - 2) dx = 1/3 x³ + 1/2 x² - 2x + C.

Find (x² + 1) (x³ - 2) dx.

Example 2. Find (2x + 1)^1/2dx.

Taking 2x + 1 = u, upon differentiating both sides we have dx = 1/2 du
and (2x + 1)^1/2dx = 1/2 u^1/2du
==> 1/2 u^3/2 / 3/2 = 1/3 (2x + 1)^3/2

Let's try an exercise problem of this type.

Find x × (x² - 1)^1/2 dx.



Example 3. Find sin (ax + b) dx.

Taking ax + b = u, we have
dx = 1/a du
and sin (ax + b)dx = 1/a sin u du.
==> -1/a cos (ax + b) + C.

Now let's try this exercise problem :

Find cos (4x - 5) dx.