Evaluate the Integral

Problem

(∫_0^π/6)(9*sec2(x)*d(x))

Solution

1. Identify the constant and move it outside the integral to simplify the expression.

(∫_0^π/6)(9*sec2(x)*d(x))=9*(∫_0^π/6)(sec2(x)*d(x))

2. Recall the fundamental trigonometric integral for the secant squared function, which is the tangent function.

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

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper and lower limits.

9*[tan(x)](π/6)0

4. Substitute the limits of integration into the antiderivative.

9*(tan(π/6)−tan(0))

5. Evaluate the trigonometric values at the specific angles.

tan(π/6)=√(,3)/3

tan(0)=0

6. Simplify the final expression by multiplying the constant by the result of the subtraction.

9*(√(,3)/3−0)=3√(,3)

Final Answer

(∫_0^π/6)(9*sec2(x)*d(x))=3√(,3)

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