Evaluate the Integral

Problem

(∫_0^π/4)(9*sec(x)*tan(x)*d(x))

Solution

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

(∫_0^π/4)(9*sec(x)*tan(x)*d(x))=9*(∫_0^π/4)(sec(x)*tan(x)*d(x))

2. Apply the fundamental theorem of calculus by finding the antiderivative of sec(x)*tan(x) which is sec(x)

9*[sec(x)](π/4)0

3. Substitute the upper and lower limits of integration into the antiderivative.

9*(sec(π/4)−sec(0))

4. Evaluate the trigonometric functions at the given points.

sec(π/4)=√(,2)

sec(0)=1

5. Simplify the expression to find the final numerical value.

9*(√(,2)−1)

9√(,2)−9

Final Answer

(∫_0^π/4)(9*sec(x)*tan(x)*d(x))=9√(,2)−9

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