Evaluate the integrals in Exercises 1–6.
∫ dt / (t - √(1...

Question

Evaluate the integrals in Exercises 1–6.

∫ dt / (t - √(1 - t²))

Accepted Answer

Evaluate the integral of dx divided by the square root of 1 + x2 + x. Now to solve this, we're going to simplify this by multiplying by the conjugate. So we have 1 divided by the square root 1 + X2 + X, multiplied by its conjugate square root 1 + X2 minus X, in our numerator and in denominator. Now this is going to simplify to the square root 1 + x2 minus x divided. By the square root 1 + x2 + x with 1 + x 2 in the square root multiplied by the square root 1 + x2. Minus X This simplifies once more to give a square root 1 minus or plus X2 minus X. So then we now have the integral square root 1 + x2 minus xdx. Which we can separate this into two integrals, the square root of 1 + X2. DX minus the integral of XDX. Now let's find our integral 1 + X2 and the square root DX. To solve this, we will use a substitution. So I'll write this over here, square root 1 + x2 d x. Where in our case, we'll let x equals sinh of u. This means d x equals cosine h u. Do you We can then rewrite this square root 1 plus sinh. You squared. Based on our identities, this simplifies to give us cosine h of u. Meaning we get the integral. Cosine HU multiplied by cosine HUDU, which is just. Cosine squared H U D U. Now, this is another one of our identities. This is the half angle identity, which means we change this to 1/2 multiplied by 1 plus cosine h. Of to you. With d u at the ends. Now we can bring out the one half to get 1/2 integral, 1 plus cosine H2UDU. We do know this antiderivative then. This is 1/2 multiplied by you. Plus, 1/2 Sign H to you. We can simplify this once more to just give us u divided by 2. + 1/2 Signh. You Cosine h u plus c. Keeping in mind That sinh to u is equivalent. To 2 sin huu cosine HU. Now we have plus C at the end, and from here, we can substitute our variables back in. We know from before that X is equals the sin HU. Which means you is equals. To sinh inverse of x. We can also note that sin hu is X from before. And cosine huu is the square root of 1 + x 2. Meaning We can add all. Of our substitutions back in We get This equals To 1/2 Natural logx plus square root. 1 + X2. + 1/2 x square root 1 + x2. Plus C. We also need to note that we had an integral from way before of XDX. The integral XDX is just X2 divided by 2 + C, so this will be tacked on to the end. We have 1/2 natural log, x + square root 1 + X2 + 1/2 x 2 root 1 + X2 + X2 divided by 2 + C. Finally, we can put everything in the same denominator of 2. Which simplifies our answer to x 2 root. 1 + x 2. Plus natural log of x + square root 1 + x 2. Minus X2 all divided by 2 + C. This is the answer to our problem. OK, I hope to help you solve it. Thank you for watching. Goodbye.

Evaluate the integrals in Exercises 1–6.∫ dt / (t - √(1 - t²))

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Evaluate the integrals in Exercises 1–6.
∫ dt / (t - √(1 - t²))