Evaluate the Integral integral of 2sec(x)^2 with respect to x

Problem

(∫_^)(2*sec2(x)*d(x))

Solution

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

(∫_^)(2*sec2(x)*d(x))=2*(∫_^)(sec2(x)*d(x))

2. Recall the basic trigonometric integral rule for the secant squared function.

d(tan(x))/d(x)=sec2(x)

3. Apply the rule to find the antiderivative of sec2(x)

(∫_^)(sec2(x)*d(x))=tan(x)+C

4. Multiply the result by the constant factor of 2

2*(tan(x)+C)=2*tan(x)+C

Final Answer

(∫_^)(2*sec2(x)*d(x))=2*tan(x)+C

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