Example 13: Find ∫ csc x dx.
You guessed it! Multiply csc x by dx. This gives you
Let u = csc x + cot x and du = (− csc x cot x − csc^2 x) dx. And, just as in Example 5, you can rewrite the
integral as
∫−
And integrate.
∫ − = −ln |u| + C = −ln |csc x + cot x| + C
Therefore, ∫csc x dx= −ln |csc x + cot x| + C.
As we do more integrals, the natural log will turn up over and over. It’s important that you get good at
recognizing when integrating requires the use of the natural log.
INTEGRATING ex AND ax
Now let’s learn how to find the integral of ex. Remember that ex = ex? Well, you should be able to
predict the following formula:
∫e
u du = eu + C