Finding Indefinite Integrals

In Exercises 17–...

Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(sin2x − csc²x)dx

Accepted Answer

Welcome back, everyone. Find the indefinite integral cosine of 2 X plus squared XDX. For this problem, let's begin by splitting the integral. We're going to get integral of cosine of 2 XDX. Plus integral of squared XD X. And now let's evaluate each integral, starting with the integral of cosine of 2 XDX. So for this integral, the main problem is that instead of X, we have another angle which is 2 X. So what we can do is simply change the differential. We can rewrite this integral as cosine of 2 X, and instead of the X, we're going to integrate with respect to 2 X. But if we change the differential, we basically have to add the fraction in front of the integral, right? Because we have 2 X, we're going to add 1/2 because when we find the derivative of 2 X, we're going to get 22 multiplied by 1/2, essentially gives us that constant of 1. So now we are ready to integrate cosine, the integral of cosine ass sign. So we're going to get one half sign. Of 2 X plus a constant of integration C1. Now for the other integral, we want to evaluate squared XD X. This is now an elementary integral and its value is tangent of X. We're going to add a constant of integration C2. So now if we substitute the results into our formula, we're going to get our final answer, which is 1/2. Sign of 2 X. Plus Now, C1 + C2 is going to result in a new constant of integration C, so we can just proceed with the next function, which is tangent of X, and then add C, where C is equal to C1 + C2, right? And that's our final answer for this problem. Thank you for watching.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫(sin2x − csc²x)dx

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Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(sin2x − csc²x)dx