Example 9: (^) ∫ cos πx dx = + C
Example 10: (^) ∫ sec tan dx = 2 sec + C
If you’re not sure if you have the correct answer when you take an integral, you can always check by
differentiating the answer and seeing if you get what you started with. Try to get in the habit of doing that
at the beginning, because it’ll help you build confidence in your ability to find integrals properly. You’ll
see that, although you can differentiate just about any expression that you’ll normally encounter, you won’t
be able to integrate many of the functions you see.
ADDITION AND SUBTRACTION
By using the rules for addition and subtraction, we can integrate most polynomials.
Example 11: Find ∫(x^3 + x^2 − x) dx.
We can break this into separate integrals, which gives us
∫^ x
(^3) dx +
∫^ x
(^2) dx −
∫^ x dx
Now you can integrate each of these individually.
You can combine the constants into one constant (it doesn’t matter how many C’s we use, because their
sum is one collective constant whose derivative is zero).
Sometimes you’ll be given information about the function you’re seeking that will enable you to solve for
the constant. Often, this is an “initial value,” which is the value of the function when the variable is zero.
As we’ve seen, normally there are an infinite number of solutions for an integral, but when we solve for
the constant, there’s only one.
Example 12: Find the equation of y where = 3x + 5 and y = 6 when x = 0.
Let’s put this in integral form.