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