Evaluate the Integral

Problem

(∫_0^π/3)(5*sec2(x)*d(x))

Solution

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

(∫_0^π/3)(5*sec2(x)*d(x))=5*(∫_0^π/3)(sec2(x)*d(x))

2. Recall the fundamental trigonometric integral formula where the antiderivative of sec2(x) is tan(x)

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

3. Apply the Fundamental Theorem of Calculus to evaluate the definite integral using the antiderivative.

5*[tan(x)](π/3)0

4. Substitute the upper and lower limits of integration into the expression.

5*(tan(π/3)−tan(0))

5. Evaluate the trigonometric values at the specific angles.

tan(π/3)=√(,3)

tan(0)=0

6. Simplify the final expression by performing the subtraction and multiplication.

5*(√(,3)−0)=5√(,3)

Final Answer

(∫_0^π/3)(5*sec2(x)*d(x))=5√(,3)

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