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Q2. Find the function G(x) that satisfies the following initial value problem? G(x) = sin(x) , =1 dx...

Question

Q2. Find the function G(x) that satisfies the following initial value problem? G(x) = sin(x) , =1 dx

Q2. Find the function G(x) that satisfies the following initial value problem? G(x) = sin(x) , =1 dx



Answers

Find each of the indefinite integrals in Problems $I-I 0 .$ $$ \int \sin 2 x d x $$

So we have ever bags is equal to two sine x plus one. And we're supposed to find Canton derivative that we do that by taking the anti derivative of To Sign X, which will be negative to co sign X and the anti girl. But of the one it's just X. And we should not forget about good policy. And we can check those by taking the derivative of the function negative too. Co sign X plus X plus e Not taking its derivative, we get to sine X visitor. But ever back this just one and a derivative of see, since it's a constant is zero. Now we'll end up with our first function.

For this question. We want to find the following integral by using use up and were given. Are you are you is equal to sign up. So that's fine. Or do you? That's people to co sign of X dx. Okay, so we see that co sign of X dx is right here. So that's equal to do you so that do you and an f applied at Sina back. And now we also have to find our our endpoints. So we know that would actually go to pi over two. We use our formula for you. You is equal to sign of pirate too. And that's equal to what? And we also have what X is equal to zero You is people to sign observable which is equal to zero. So we have our endpoints are from 0 to 1. Okay, so we get the following integral

In problem thing. We want to find the indefinite integral of X plus form multiplied by sign X squared plus two. X. The X. We can notice here the differentiation of the inner function of science is multiplied by the sine here, all part of it because the differentiation is two X plus two. And when we take two as a common factor it's two multiplied by X plus one and X plus one is here. Then we can use a trendy geometric substitution of U equals what Inside the sine function U equals X squared plus two X. And we get you equals two X plus two. Then we use this substitution to find the integral. It equips the integral of X plus one, multiplied by sine X squared plus two. X. Dx. We can substitute by X plus one. Multiply by dx to be. Do you divided by two? If here were divided by two we get X plus one and we hear forget the X. It's X plus one. The X. Then X plus one. The X will be substituted by D. You divided by two. Do you divided by two? And we have signed? We substitute X squared plus two X by you. Now this integration is very easy to integrate and find equals half the integration of sinew is minus because I knew and we add the constant because this is indefinite integral. We should get back to X. Then we remove you and substitute by X squared plus two X instead of equals minus Hoff design of X squared plus two X plus the. And this is a final answer of our problem.

From the given in Dingle we make this of Tillerson. You Would you goto cost the works? Differentiating both sides were d which is equal to minus two. Signed two x So we have minus do you to do you goto signed two x dx so we can identical s minus one by two until the snow u to the power you do you What do you call two minus one by two? Until this integrated the power you wake up to the power of you. Let's see So it's a starting back. We have minus one by two. It was a hard core two weeks plus c.


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