$\int_0^{\frac{\pi}{3}} \frac{d x}{\cos x}$
Answer (2)
The definite integral ∫ 0 3 π cos x d x simplifies to ∫ 0 3 π sec x d x , which evaluates to ln ( 2 + 3 ) . To evaluate, we recognize that at the limits x = 3 π and x = 0 , we find the values of sec and tan and substitute them into the logarithmic expression. The final result is ln ( 2 + 3 ) .
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Rewrite the integrand as sec x .
Integrate sec x to get ln ∣ sec x + tan x ∣ .
Evaluate the definite integral at the limits 3 π and 0 .
Simplify the expression to obtain the final answer: ln ( 2 + 3 ) .
Explanation
Rewrite the integral We are asked to evaluate the definite integral ∫ 0 3 π cos x d x First, we recognize that c o s x 1 = sec x . Thus, we need to evaluate ∫ 0 3 π sec x d x
Integrate sec(x) To integrate sec x , we multiply by s e c x + t a n x s e c x + t a n x :
∫ sec x d x = ∫ sec x ⋅ sec x + tan x sec x + tan x d x = ∫ sec x + tan x sec 2 x + sec x tan x d x Notice that the derivative of the denominator is the numerator, i.e., d x d ( sec x + tan x ) = sec x tan x + sec 2 x . Therefore, the integral becomes ∫ sec x + tan x sec 2 x + sec x tan x d x = ln ∣ sec x + tan x ∣ + C
Evaluate the definite integral Now we evaluate the definite integral: ∫ 0 3 π sec x d x = ln ∣ sec x + tan x ∣ 0 3 π = ln sec ( 3 π ) + tan ( 3 π ) − ln ∣ sec ( 0 ) + tan ( 0 ) ∣ We know that sec ( 3 π ) = 2 , tan ( 3 π ) = 3 , sec ( 0 ) = 1 , and tan ( 0 ) = 0 . Substituting these values, we get ln ∣2 + 3 ∣ − ln ∣1 + 0∣ = ln ( 2 + 3 ) − ln ( 1 ) = ln ( 2 + 3 ) − 0 = ln ( 2 + 3 )
Final Answer Therefore, the final answer is ln ( 2 + 3 ) . We can approximate this value using a calculator: ln ( 2 + 3 ) ≈ 1.31695789692
Conclusion The definite integral ∫ 0 3 π c o s x d x evaluates to ln ( 2 + 3 ) .
Examples
Imagine you're designing a solar panel that needs to capture sunlight efficiently. The amount of sunlight hitting the panel at different angles can be modeled using trigonometric functions. Evaluating integrals involving trigonometric functions, like the one we just solved, helps you calculate the total solar energy captured over a specific period. This is crucial for optimizing the panel's design and maximizing its energy output. Understanding these concepts allows you to create more efficient and sustainable energy solutions.
