Evaluate the Integral

Problem

(∫_^)(5*sec(x)*tan(x)*d(x))

Solution

1. Identify the constant factor in the integrand and move it outside the integral.

(∫_^)(5*sec(x)*tan(x)*d(x))=5*(∫_^)(sec(x)*tan(x)*d(x))

2. Recall the basic trigonometric derivative rule where the derivative of sec(x) is sec(x)*tan(x)

d(sec(x))/d(x)=sec(x)*tan(x)

3. Apply the power of the antiderivative rule to evaluate the integral.

(∫_^)(sec(x)*tan(x)*d(x))=sec(x)+C

4. Multiply the result by the constant factor and include the constant of integration C

5*(sec(x)+C)=5*sec(x)+C

Final Answer

(∫_^)(5*sec(x)*tan(x)*d(x))=5*sec(x)+C

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