Evaluate the integrals in Exercises 33–54.
49. ∫ e^(sec ...

Question

Evaluate the integrals in Exercises 33–54.

49. ∫ e^(sec πt) sec πt tan πt dt

Accepted Answer

Welcome back everyone. Evaluate the integral of e to the power of tangent 2 x multiplied by sequin squared 2 x dx. We're going to approach this problem using u substitution. Let's suppose that u is equal to tangent of 2 x, and let's remember that our next step is to find the derivative of u with respect to x. That's the derivative of tangent of 2 x. We can find that using the chain rule. So first of all, we're going to get sequin squared of 2 x, and we want to multiply by the derivative of 2x, which is 2. Now in the integral we have sequin 22 x dx, so we can show that sequin square of 2 x multiplied by dx is simply duu divided by 2 or 1/2 duu. And now this allows us to rewrite the integral in terms of u and duu. So we get e to the power of u multiplied by sequin square of 2 x dx, which is 1/2 duu. We can write duu and we can factor out 1/2. And now we have an elementary integral. We're going to write one half, which is our constant multiplied by the integral of e to the power of u d u, which is e to the power of u, plus a constant of integration c. Our final step is to replace u with tangent of 2 x, which gives us one half e to the power of tangent of 2 x. Plus a constant of integration c. This is our final answer, and thank you for watching.

Evaluate the integrals in Exercises 33–54.49. ∫ e^(sec πt) sec πt tan πt dt

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Evaluate the integrals in Exercises 33–54.
49. ∫ e^(sec πt) sec πt tan πt dt