Evaluating integrals Evaluate the following integrals.

∫(√1 + tan 2t) sec² 2t dt

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves a composite function. The term √(1 + tan(2t)) suggests a substitution might simplify the integral. Let u = 1 + tan(2t).

Step 2: Differentiate u with respect to t. Since u = 1 + tan(2t), we have du/dt = 2 sec²(2t). Therefore, du = 2 sec²(2t) dt.

Step 3: Rewrite the integral in terms of u. Substitute u = 1 + tan(2t) and du = 2 sec²(2t) dt into the integral. This gives (1/2) ∫√u du.

Step 4: Solve the simplified integral. The integral of √u is (2/3)u^(3/2). So, (1/2) ∫√u du becomes (1/3)u^(3/2) + C, where C is the constant of integration.

Step 5: Substitute back u = 1 + tan(2t) to return to the original variable. The final expression is (1/3)(1 + tan(2t))^(3/2) + C.

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