Video Transcript
Determine the indefinite integral
of cot 𝑥 with respect to 𝑥.
Now we know that cot 𝑥 can be
written as cos of 𝑥 over sin of 𝑥. Therefore, we can rewrite our
integral as the integral of cos 𝑥 over sin 𝑥 with respect to 𝑥. Next, we’ll be using the fact that
the differential of sin 𝑥 with respect to 𝑥 is equal to cos 𝑥. And so if we let our denominator
sin 𝑥 be equal to 𝑓 of 𝑥, then our numerator, cos of 𝑥, will be equal to 𝑓
prime of 𝑥. And here, we can see that our
integral is of the form of the integral of 𝑓 prime of 𝑥 over 𝑓 of 𝑥 with respect
to 𝑥.
And we know a formula for solving
integrals of this form. And this formula tells us that the
indefinite integral of 𝑓 prime of 𝑥 over 𝑓 of 𝑥 with respect to 𝑥 is equal to
the natural logarithm of the absolute value of 𝑓 of 𝑥 plus 𝑐. If we apply this formula to our
integral with 𝑓 of 𝑥 is equal to sin 𝑥, then we will reach our solution. Which is that the indefinite
integral of cot 𝑥 with respect to 𝑥 is equal to the natural logarithm of the
absolute value of sin 𝑥 plus 𝑐.