Evaluating integrals Evaluate the following integrals.

∫π/₆^π/³ (sec² t + csc² t) dt

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves the sum of two trigonometric functions: sec²(t) and csc²(t). Recall their respective antiderivatives: ∫sec²(t) dt = tan(t) and ∫csc²(t) dt = -cot(t).

Step 2: Break the integral into two separate integrals: ∫(sec²(t) + csc²(t)) dt = ∫sec²(t) dt + ∫csc²(t) dt.

Step 3: Compute the antiderivative of each term. For sec²(t), the antiderivative is tan(t). For csc²(t), the antiderivative is -cot(t). Combine these results to get tan(t) - cot(t).

Step 4: Apply the Fundamental Theorem of Calculus to evaluate the definite integral. Substitute the upper limit π/3 and the lower limit π/6 into the antiderivative expression tan(t) - cot(t).

Step 5: Simplify the result by calculating tan(π/3), tan(π/6), cot(π/3), and cot(π/6). Subtract the values obtained at the lower limit from those at the upper limit to find the final value of the integral.

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