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.

∫ (sec t + cot t)² dt

Accepted Answer

Welcome back everyone. Evaluate the integral of cosequin c plus cotangent c squad dc. For this problem, let's go ahead and square the sum. So we are integrating co sequin squared of t + 2 co sequinc. Cotangent C+ cotangent squatdt. So now what we can do is simply split this integral into three parts. That would be integral of cosse and square of TDT plus 2 integral of cosequ and C. Cootent. TDT. Plus integral of co tension squared of cdc. And now we can notice that each integral is an elementary integral. Additionally, we can express cotent square of t in terms of co sequin squared c minus 1, which will basically show that we can use elementary integrals to evaluate it, right? So now let's integrate each step by step, starting with co sequin squared TDT. The value of this integral is negative cotangent c plus a constant of integration. Let's call it C1. Now the integral of coequantt cotangent t dt is equal to. Negative cosequent t. Plus a constant of integration, let's call it C2. And as we mentioned, the integral of cotangent squared of cdc can be evaluated by using trigonometric identity. Cotangent squared is cosein squared. T minus 1, right? So the value of this integral is going to be equal to negative cotangents, negative cotangentc. first of all, as we know, that's the integral of co sequin squared of c, so we get negative cotangent of t. And then the integral of -1 is going to be net plus a constant of integration. Let's call it c3. So now when we combine these, we are going to get. The value of the original integral, let's call that integral i. So the value of i is going to be equal to negative cotangent. See Plus C1. Then we are adding 2 multiplied by negative cosequinc, so we're subtracting 2 cosequinc. And adding 2C2. And then we are adding integral of cotangent squared of c, which is basically negative cotangent c minus c. Plus C 3 Now, let's go ahead and simplify. Notice that we have negative cotangent minus cotangents, so that's -2 cotangent T. Then we have 2 cosequent C, so we're going to subtract 2 cosequent c. And we also have negative T. For all the constants we can combine them and we can just say plus c. Well done. So that's our final answer. Let's label it and 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.∫ (sec t + cot t)² dt

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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.
∫ (sec t + cot t)² dt