Integrating csc(x) cot(x) using substitution: Step-by-step guide

cscx cotx dx

To integrate the expression csc(x) cot(x) dx, we will use a technique called substitution.

Step 1: Start by rewriting the expression using trigonometric identities.

Recall that:

csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x).

Substituting these values into the expression, we have:

csc(x) cot(x) dx = (1/sin(x)) * (cos(x)/sin(x)) dx

Step 2: Simplify the expression further.

Multiplying the terms together, we get:

(1/sin(x)) * (cos(x)/sin(x)) dx = cos(x)/sin^2(x) dx

Step 3: Substitute u = sin(x).

Let’s make the substitution: u = sin(x). This will simplify the expression and make it easier to integrate.

Taking the derivative of both sides of the equation, we have:

du/dx = cos(x)

Rearranging the equation, we can express cos(x) in terms of u:

cos(x) = du/dx

Substituting this into our expression:

cos(x)/sin^2(x) dx = (du/dx)/sin^2(x) dx = du/sin^2(x)

Step 4: Rewrite the expression in terms of u.

We need to express sin^2(x) in terms of u. Using the identity sin^2(x) + cos^2(x) = 1, we can rearrange it to:

sin^2(x) = 1 – cos^2(x)

Substituting cos^2(x) with 1 – sin^2(x), we have:

sin^2(x) = 1 – (1 – u^2) = u^2

So, du/sin^2(x) = du/u^2 = u^(-2) du

Step 5: Integrate the expression.

Now, we can integrate the expression:

∫ u^(-2) du = -u^(-1) + C

Step 6: Substitute back in terms of x.

Finally, substituting u = sin(x) back into the expression, we get:

∫ csc(x) cot(x) dx = -(sin(x))^(-1) + C

Or, simplifying it:

∫ csc(x) cot(x) dx = -cosec(x) + C

Therefore, the integral of csc(x) cot(x) dx is equal to -cosec(x) + C, where C is the constant of integration.