The integrals in Exercises 1–44 are in no particular order. Ev...

Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (dθ / cos θ - 1)

Accepted Answer

Welcome back everyone. Evaluated the integral dc divided by 1 plus sinc. So for this problem we're going to rewrite the integral. That's the integral of d c divided by 1 plus sinc, and one of the ways to solve this integral is to multiply both sides of the fraction by the conjugate of 1 plus sin t, which is 1 minus sinc. And we are going to divide it by 1 minus sinet, so the integral remains unchanged. And now we can rewrite it as 1 minus sinc. DC Divided by 1 minus sin squat, right, because we have a factored form of the difference of squares. And now what is 1 minus sin squared c? Well, according to the Pythagorean identity, that's cosine squared c. So we can just rewrite the denominator as Cosine squared t and we can now split our integral into two parts so we're going to have integral of. 1 divided by cosine squared cdc. minus integral of Sint divided by Cosine squared cdc. And now let's go ahead and rewrite each integral as follows. For the first integral, 1 divided by cosine squared is sequin 2, so we get the integral of sequin 2 cdc. And for the 2nd integral, we have sin c divided by cosine squared c. It can be written as sin c divided by. Cosinec multiplied by 1 divided by cosinec. So the first term is tangent c and the second one is 2. See, so we can rewrite this expression as Ancient Of tea Multiplied by 2 of cdc. Well done, so now let's integrate. Starting with the integral of sequin square cdc, this is an elementary integral and this value is tangent c. And for the second integral again we have an elementary integral. Integral tangent c sequent cdc is equal to sequent c. So we're going to subtract sequent c and we're going to add the constant of integration c because we have indefinite integrals. Thank you for watching.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ (dθ / cos θ - 1)

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The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dθ / cos θ - 1)