Question

∫ex (1 - cotx + cot2x) dx = ?

Answer 1

Step 1: Simplify the Integrand using Trigonometric Identities
We are given the integral ∫ eˣ (1 - cot(x) + cot²(x)) dx.
Recall that cot²(x) = csc²(x) - 1. Substituting this into the integral, we get:
∫ eˣ (1 - cot(x) + csc²(x) - 1) dx = ∫ eˣ (csc²(x) - cot(x)) dx.

Step 2: Recognize the Derivative Relationship
We know that the derivative of cot(x) with respect to x is -csc²(x).
Let f(x) = cot(x). Then f'(x) = -csc²(x). Therefore, csc²(x) = -f'(x).

Step 3: Rewrite the Integral in Terms of f(x) and f'(x)
Substitute f(x) and f'(x) into the integral:
∫ eˣ (csc²(x) - cot(x)) dx = ∫ eˣ (-f'(x) - f(x)) dx = -∫ eˣ (f(x) + f'(x)) dx.

Step 4: Apply the Integration Rule for eˣ[f(x) + f'(x)]
The integration rule states that ∫ eˣ (f(x) + f'(x)) dx = eˣ f(x) + C, where C is the constant of integration.
Since we have -∫ eˣ (f(x) + f'(x)) dx, the result will be -eˣ f(x) + C.

Step 5: Substitute f(x) = cot(x) to Obtain the Final Result
Substitute f(x) = cot(x) back into the expression:
-eˣ f(x) + C = -eˣ cot(x) + C.

Therefore, the integral ∫ eˣ (1 - cot(x) + cot²(x)) dx = -eˣ cot(x) + C.