Example 1: Solve integral of exponential function ∫e^{x3}2x^{3}dx
Solution:
Step 1: the given function is ∫e^{x}^^{3}3x^{2}dx
Step 2: Let u = x^{3} and du = 3x^{2}dx
Step 3: Now we have: ∫e^{x}^^{3}3x^{2}dx= ∫e^{u}du
Step 4: According to the properties listed above: ∫e^{x}dx = e^{x}+c, therefore ∫e^{u}du = e^{u} + c
Step 5: Since u = x^{3} we now have ∫e^{u}du = ∫e^{x3}dx = e^{x}^^{3} + c
So the answer is e^{x}^^{3} + c
Example 2: Integrate .
Solution: First, split the function into two parts, so that we get:
Example 3: Integrate ∫lnx dx.
Solution:
Let
u=lnu
and
dv = dx = (1)dx
so that:
du = 1/x dx
and
v = x
Therefore:
∫lnx dx
= x lnx - ∫x * 1/x dx
=xlnx - ∫1dx
=xlnx – x + C
Example 4: Integrate .
Solution:
Use u-substitution. Let u = 3 + lnx
So that du = 1/x dx.
Substitute into the original problem we will get: