Evaluate the Integral integral of sec(x) with respect to x

Problem

(∫_^)(sec(x)*d(x))

Solution

1. Multiply the integrand by a form of 1 that will facilitate substitution, specifically (sec(x)+tan(x))/(sec(x)+tan(x))

(∫_^)(sec(x)⋅(sec(x)+tan(x))/(sec(x)+tan(x))*d(x))

2. Distribute the sec(x) in the numerator.

(∫_^)((sec2(x)+sec(x)*tan(x))/(sec(x)+tan(x))*d(x))

3. Identify a substitution where u=sec(x)+tan(x)

u=sec(x)+tan(x)

4. Differentiate u with respect to x to find d(u)

d(u)=(sec(x)*tan(x)+sec2(x))*d(x)

5. Substitute u and d(u) into the integral.

(∫_^)(1/u*d(u))

6. Integrate with respect to u

ln(u)+C

7. Substitute back the original expression for u

ln(sec(x)+tan(x))+C

Final Answer

(∫_^)(sec(x)*d(x))=ln(sec(x)+tan(x))+C

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