Evaluate the Integral integral of cot(x)^3 with respect to x

Problem

(∫_^)(cot3(x)*d(x))

Solution

1. Use a trigonometric identity to rewrite the integrand by splitting cot3(x) into cot(x)*cot2(x)

(∫_^)(cot(x)*cot2(x)*d(x))

2. Substitute the identity cot2(x)=csc2(x)−1 into the integral.

(∫_^)(cot(x)*(csc2(x)−1)*d(x))

3. Distribute the cot(x) term to create two separate integrals.

(∫_^)(cot(x)*csc2(x)*d(x))−(∫_^)(cot(x)*d(x))

4. Apply u-substitution to the first integral by letting u=cot(x) which means d(u)=−csc2(x)*d(x)

(∫_^)(−u*d(u))−(∫_^)(cot(x)*d(x))

5. Integrate both terms using the power rule for u and the standard integral formula for cot(x)

−(u2)/2−ln(sin(x))+C

6. Substitute back u=cot(x) to express the final result in terms of x

−cot2(x)/2−ln(sin(x))+C

Final Answer

(∫_^)(cot3(x)*d(x))=−cot2(x)/2−ln(sin(x))+C

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