What's the integral of csc(x)cot(x)?

The integral of csc(x)cot(x) is -csc(x) + C.

To find the integral of csc(x)cot(x), we can use the substitution u = csc(x), du/dx = -csc(x)cot(x) dx. Therefore, the integral becomes:

∫csc(x)cot(x) dx = -∫du

Using the trigonometric identity csc(x) = 1/sin(x), we can rewrite the integral as:

∫csc(x)cot(x) dx = ∫(cos(x)/sin^2(x)) dx

Using the substitution u = sin(x), du/dx = cos(x), the integral becomes:

∫(cos(x)/sin^2(x)) dx = -∫du/u^2

Integrating this expression gives:

∫du/u^2 = -u^-1 + C = -csc(x) + C

Therefore, the final answer is:

∫csc(x)cot(x) dx = -csc(x) + C

To understand the trigonometric identities used in this integral, refer to our detailed discussion on trigonometric identities. Additionally, for more complex integrals involving trigonometric functions, exploring our page on integration of trigonometric functions can provide deeper insights. If you are interested in different techniques to approach integration, our section on techniques of integration might be helpful.

A-Level Maths Tutor Summary: To solve the integral of csc(x)cot(x), we substitute csc(x) for u, making the integral simpler. We then use trig identities to express the integral in terms of u, and integrate to find -csc(x) + C. This method involves recognising patterns and applying substitutions to simplify the integral, a common technique in calculus that's useful for solving integrals more easily.