7–84. Evaluate the following integrals.
79. ∫ (arcsinx)/...

Question

7–84. Evaluate the following integrals.

79. ∫ (arcsinx)/x² dx

Accepted Answer

Hello. In this video, we are going to be evaluating the following definite integral. So we are given the integral of tangent inverse of X divided by X squared. Now, we can approach this integral by using a U substitution. Let's go ahead and allow you to be equal to the problematic term here, which is going to be tangent inverse of X. Now, it's going to be difficult to find a substitution for DX, so let's go ahead and solve for X in this case. If we were to solve for X, we will end up with X's equal. To tangent Of you And by taking the derivative of both sides of this equation, we end up with DU is equal to C or DX is equal to Cu DU. Let's go ahead and apply these substitutions to the original integral. That is going to allow us to rewrite the original integral as the integral of U divided by X2, which is tangent squared of u multiplied by sequent squared of U D U. Now, seek and square of view divided by tangent squared of view is going to simplify to be cossein squared of you. So we have the integral of U multiplied by cosequent squared of U D U. Now, what we're going to do here is we're going to approach this problem by using integration by parts. Now, we are going, normally integration by parts is in the terms of UMV, but we're going to save the confusion and use A and B instead. So we're going to write the integral of the form ADB, and this is going to be equal to A multiplied by B minus the integral of BDA. So here we're going to allow our A term to equal to you. And we're going to allow DB to equal to cosin squared of you. By taking the derivative of our A term, we get DA is equal to DU, and by integrating our DB term, we get B is equal to negative cotangent of U. And by using the formula for the integration by parts, we can rewrite the given integral. As a multiplied by B, which is negative U multiplied by cotangent of you. Minus the integral of negative cotangent of you. Do you Now what we're going to do is simplify. We can make this integral positive by pulling out the negative term. So we will have negative U multiplied by cotangent of U, plus the integral of cotenent of U D U. Now, by the table of integrals, the identity for the integral of cotangent of you. is equal to the natural logarithm. Of sign of you So that last integral is going to evaluate, and that is going to simplify this solution to be negative U multiplied by cotangent of U, plus the natural logarithm of sine of you. And because we were working with an indefinite integral, we will add a C term. Now, what we will go ahead and do is we will go ahead and back substitute. All the Us in terms of X. Now that was going to give us the first term negative view, which is going to be negative, tangent inverse of X. Now for cotangent of you, we know that from this second statement, X is equal to tangent of you, but cotangent is the inverse of the tangent value, and that means that cotangent of view can be rewritten as 1 divided by X. So cotangent of view can be substituted to be one divided by X. And We will go ahead and. Substitute sign of you by using a right triangle for tangent of you. Now we know that tangent of you. is equal to x and by this right triangle. This tells us that we have X on the opposite end of theta, and we have 1 for the adjacent side of theta. Now, if we were to solve our missing side, we get B2 is equal to X2 + 1, which is going to be the square root of X2 + 1. So we know that this hypotenuse will be X2 + 1, and if we were to solve for sin of you in this case, we know that sine of u is going to equal to X divided by the square root. of X + X2 + 1 or 1 + X squad. And so that means that we can substitute the sign of you to be the natural logarithm of X divided by the square root of 1 + X2 + C. And finally, if we were to reorganize the terms and simplify, the final solution is going to be the natural logarithm of X divided by the square root of 1 + X2. Minus tangent inverse of X divided by X plus C, and this is going to be the final solution to the integral. And with that being said, the overall solution is going to be B. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

7–84. Evaluate the following integrals.
79. ∫ (arcsinx)/x² dx

Key Concepts

Integration

Arcsine Function

Improper Integrals

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