Evaluate the Integral

Problem

(∫_0^π/4)(7*sec(x)*tan(x)*d(x))

Solution

1. Factor out the constant from the integral to simplify the expression.

(∫_0^π/4)(7*sec(x)*tan(x)*d(x))=7*(∫_0^π/4)(sec(x)*tan(x)*d(x))

2. Identify the antiderivative of sec(x)*tan(x) which is sec(x)

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

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

7*[sec(x)](π/4)0

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

7*(sec(π/4)−sec(0))

5. Evaluate the trigonometric functions at the given points.

sec(π/4)=√(,2)

sec(0)=1

6. Simplify the final expression by distributing the constant.

7*(√(,2)−1)

Final Answer

(∫_0^π/4)(7*sec(x)*tan(x)*d(x))=7√(,2)−7

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