Evaluate the Integral

Problem

(∫_^)(sec(x)/tan(x)*d(x))

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

(∫_^)(1/cos(x)/sin(x)/cos(x)*d(x))

2. Simplify the fraction by multiplying the numerator by the reciprocal of the denominator.

(∫_^)(1/cos(x)⋅cos(x)/sin(x)*d(x))

3. Cancel the common factor of cos(x) from the numerator and denominator.

(∫_^)(1/sin(x)*d(x))

4. Identify the reciprocal identity 1/sin(x)=csc(x)

(∫_^)(csc(x)*d(x))

5. Apply the standard integral formula for the cosecant function.

−ln(csc(x)+cot(x))+C

Final Answer

(∫_^)(sec(x)/tan(x)*d(x))=−ln(csc(x)+cot(x))+C

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