Evaluate the Integral

Problem

(∫_^)(cos(x)/sin2(x)*d(x))

Solution

1. Identify a suitable substitution by noticing that the numerator cos(x) is the derivative of the base of the power in the denominator sin(x)

2. Substitute u=sin(x) which implies that d(u)=cos(x)*d(x)

3. Rewrite the integral in terms of u

(∫_^)(1/(u2)*d(u))

4. Apply the power rule for integration by expressing the integrand as u(−2)

(∫_^)(u(−2)*d(u))=(u(−1))/(−1)+C

5. Simplify the resulting expression:

−1/u+C

6. Back-substitute u=sin(x) to return to the original variable:

−1/sin(x)+C

7. Use the trigonometric identity csc(x)=1/sin(x) to write the final form.

Final Answer

(∫_^)(cos(x)/sin2(x)*d(x))=−csc(x)+C

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