Question Video: Evaluating Indefinite Integrals Mathematics • Third Year of Secondary School

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.