Use the table of integrals at the back of the text to evaluate...

Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ tan^(-1)(x) / x² dx

Accepted Answer

Hello there. Today we are going to solve the following practice from together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Evaluate the integral, the integral of inverse sin of x divided by x to the power of 2 dx. OK. So it appears for this particular problem, we're ultimately asked to evaluate the integral that is provided to us. So now that we know that we're trying to solve for this integral. We need to recall and note based on our prior knowledge that in order to evaluate the integral. Of Inverse sign of X. Divided by X to the power of 2 DX, we can recall and use the standard formula. So with that said, We need to recall and note that the standard formula approach states that for an integral of the form, the integral of x to the power of n multiplied by inverse sign of a x dx, the integration by parts reduction formula will state that the integral of X to the power of n multiplied by inverse sign of a x dx will be equal to x to the power of n + 1 divided by n + 1 multiplied by inverse sin of a x minus a divided by n + 1. Multiplied by the interval of x to the n + 1 divided by the square root of 1 minus a 2 multiplied by x 2. DX. Fantastic. So when we apply the formula, we need to note that for our specific problem, we know that N will be equal to -2, since, let's make a note, since the term is 1 divided by x 2. And we will also know that a will be equal to 1 for this particular problem. And when we substitute these values back into our integral, we will have the integral of x to the power of -2 multiplied by inverse sin of x dx, which will be equal to x -2 + 1 divided by -2 + 1. Multiplied by inverse sin of x minus 1 divided by -2 + 1 multiplied by the interval of x to the power of -2 + 1 divided by the square root of 1 minus x 2. And don't forget DX. OK. So, moving right along here, we now need to simplify the coefficients. So we need to go ahead and write that the interval Of Inverse sign of X so inverse. Sin of x divided by x to the power of 2. D X will be equal to -1 divided by x multiplied by inverse sin of x plus the interval of 1 divided by x multiplied by the square root of 1 minus x 2. D X And now, at this point, we need to solve for the resulting integral, and we need to note that the integral of 1 divided by x multiplied by the square root of 1 minus x 2 DX is a standard form, so that will mean that its standard evaluation is. Going to be equal to the negative natural log of the absolute value of 1 plus the square root of 1 minus x 2. Divided by x plus C, where C denotes some constant integer, so once again that will mean that our integral of 1 divided by x multiplied by the square root of 1 minus x to the 2 dx. Will be equal to negative natural log of the absolute value of 1 plus the square root of 1 minus x 2 all divided by x plus c. So, when we combine the like terms or when we combine our terms, the final evaluated integral will be the integral of inverse sin of x divided by x to the power of 2 dx will be equal to negative inverse sin of x divided by x. Minus the natural log of the absolute value of 1 plus the square root of 1. Minus x 2. All divided by X. Plus C. And that's it. That is our final answer. Hooray, we did it. So, with that said, Let us quickly box our answers so we remember what our final answer is. And then you need to scroll up to the top to recap what we've learned. So, once again, we've officially solved for our final problem, our final answer for this particular prompt. So we can safely say once again our final answer without a doubt is as follows. Negative inverse sign of x divided by x minus the natural log of the absolute value of 1 plus the square root of 1 minus x to the power of 2 divided by x plus c, and that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ tan^(-1)(x) / x² dx

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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx