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

Problem

(∫_^)(csc3(x)*d(x))

Solution

1. Apply integration by parts by setting u=csc(x) and d(v)=csc2(x)*d(x)

2. Differentiate and integrate to find d(u)=−csc(x)*cot(x)*d(x) and v=−cot(x)

3. Substitute into the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

(∫_^)(csc3(x)*d(x))=−csc(x)*cot(x)−(∫_^)((−cot(x))*(−csc(x)*cot(x))*d(x))

4. Simplify the integral on the right side.

(∫_^)(csc3(x)*d(x))=−csc(x)*cot(x)−(∫_^)(csc(x)*cot2(x)*d(x))

5. Use the trigonometric identity cot2(x)=csc2(x)−1

(∫_^)(csc3(x)*d(x))=−csc(x)*cot(x)−(∫_^)(csc(x)*(csc2(x)−1)*d(x))

6. Distribute and split the integral.

(∫_^)(csc3(x)*d(x))=−csc(x)*cot(x)−(∫_^)(csc3(x)*d(x))+(∫_^)(csc(x)*d(x))

7. Add the integral (∫_^)(csc3(x)*d(x)) to both sides to group like terms.

2*(∫_^)(csc3(x)*d(x))=−csc(x)*cot(x)+(∫_^)(csc(x)*d(x))

8. Evaluate the integral of csc(x) which is −ln(csc(x)+cot(x))

2*(∫_^)(csc3(x)*d(x))=−csc(x)*cot(x)−ln(csc(x)+cot(x))

9. Divide by 2 and add the constant of integration C

(∫_^)(csc3(x)*d(x))=−1/2*csc(x)*cot(x)−1/2*ln(csc(x)+cot(x))+C

Final Answer

(∫_^)(csc3(x)*d(x))=−1/2*csc(x)*cot(x)−1/2*ln(csc(x)+cot(x))+C

Want more problems? Check here!