Verify the integration formulas in Exercises 37–40.
39. ...

Question

Verify the integration formulas in Exercises 37–40.

39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C

Accepted Answer

Evaluate the integral of X10H to the -1 XDX. Now, to solve this, we're going to use integration by parts, which tells us that if we have the integral. UDV is equivalent to UV minus the integral VDU. So in our case, we have to define U and V. We'll say you Is the 10 each in verse? And DU is a derivative of this. Which is just 1 divided by 1 minus X2. DV Is XDX, which makes V X 2 divided by 2. That'll supply our formula. We have the integral UDV. This turns into UV or X2 divided by 2, TNH inverse X. Minus the integral of VDU. X 2 divided by 2. Multiplied by 1 divided by 1 minus X2 DX. Now we have to solve our other integral. can actually simplify this part on the right here. And we ended up getting X2 divided by 1 minus X2DX. We do still have that 2 in the denominator, which we'll just bring out. To be 1/2. Now, we can simplify this integrand. We know that this, by algebra, turns into the following. -1 + 1 divided by 1 minus X2. We'll put DX at the end. Now we just need to find this integral. Now, the integral of 1 divided by 1 minus X2 is actually equivalent to tangent H inverse X. This is a standard antiderivative. That means then This turns into -1/2 multiplied. Bye Negative X. Plus 10 H in verse. A X. We can then Simplify this. We get X divided by 2 minus 10H inverse X. Divided by 2 And we can add this onto our initial term, X2 divided by 2, 10H inverse X. Finally, from here, we can simplify this. We end up getting 1/2 multiplied. By x2 minus 1. 10 h inverse of x. Plus x And then finally, we can add a plus C at the ends. This is the final answer to our problem. OK, I hope to help you solve it. Thank you for watching. Goodbye.

Verify the integration formulas in Exercises 37–40.39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C

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Verify the integration formulas in Exercises 37–40.
39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C