Evaluate the Integral

Problem

(∫_^)(1/(1−sin(x))*d(x))

Solution

1. Multiply the numerator and the denominator by the conjugate of the denominator, which is 1+sin(x)

(∫_^)((1+sin(x))/((1−sin(x))*(1+sin(x)))*d(x))

2. Simplify the denominator using the difference of squares identity.

(∫_^)((1+sin(x))/(1−sin^2(x))*d(x))

3. Apply the Pythagorean identity 1−sin^2(x)=cos^2(x) to the denominator.

(∫_^)((1+sin(x))/cos^2(x)*d(x))

4. Split the integral into two separate fractions.

(∫_^)(1/cos^2(x)+sin(x)/cos^2(x)*d(x))

5. Rewrite the trigonometric terms using basic identities 1/cos^2(x)=sec^2(x) and sin(x)/cos^2(x)=sec(x)*tan(x)

(∫_^)(sec^2(x)+sec(x)*tan(x)*d(x))

6. Integrate each term using standard integration rules.

tan(x)+sec(x)+C

Final Answer

(∫_^)(1/(1−sin(x))*d(x))=tan(x)+sec(x)+C

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