Evaluate the Integral

Problem

(∫_^)(sec2(x)−sin(x)*d(x))

Solution

1. Apply the sum rule for integration to split the integral into two separate parts.

(∫_^)(sec2(x)−sin(x)*d(x))=(∫_^)(sec2(x)*d(x))−(∫_^)(sin(x)*d(x))

2. Evaluate the first integral using the standard trigonometric integral rule where the derivative of tan(x) is sec2(x)

(∫_^)(sec2(x)*d(x))=tan(x)

3. Evaluate the second integral using the rule where the derivative of cos(x) is −sin(x)

(∫_^)(sin(x)*d(x))=−cos(x)

4. Combine the results and add the constant of integration C

tan(x)−(−cos(x))+C

5. Simplify the signs to reach the final expression.

tan(x)+cos(x)+C

Final Answer

(∫_^)(sec2(x)−sin(x)*d(x))=tan(x)+cos(x)+C

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