92–98. Evaluate the following integrals.
97. ∫ tan⁻¹(∛x)...

Question

92–98. Evaluate the following integrals.

97. ∫ tan⁻¹(∛x) dx

Accepted Answer

Hello. In this video, we are going to be evaluating the indefinite integral. We are giving the integral of tangent inverse of the square root of X. Now, we can approach this problem by using a U substitution. If we let u equal to tangent inverse of the square root of X. Then we can solve for X in this case, and we will get that X is equal. To tangent squared of you. And if we were to take the derivative of both sides of the statement, we will end up with DX is equal to 2 multiplied by tangent of U, multiplied by C squad of U D U. And by applying these substitutions to the integral, we can rewrite the integral to be the integral of you multiplied by 2, which will which will move to the front of the integral as a constant. Multiplied by tangent of U, multiply by seek and square of U D U. Now, we can approach this part of the problem by using integration by parts. We are going to have an integral of the form A multiplied by DB, and this is going to be equivalent to A multiplied by B minus the integral of B multiplied by DA. So we need to define our A and B terms in this case. We are going to allow A to equal to U, and we're going to allow DB to equal to tangent of U multiplied by skin square of U D U. By taking the derivative of our A term, we will end up with DA is equal to DU, and by integrating our DB term, we will have B is equal to 1/2 tangent squared of U. And that is going to be all the terms we need for integration by parts. So, by applying the integration by parts, we will have the coefficient of 2 multiplied by A multiplied by B, which is 1/2 U multiplied by tangent square of U. Minus The integral Of one half tangent squared of you. Do you Now, we can simplify this by first distributing the two coefficient to everything in the integration by parts. That will simplify the integration by parts to be U multiplied by tangent squared of you, minus the integral of tangent squared of you. Do you Now, we can simplify tangent squared of you by using our trigonometric identities. We can rewrite the integral to be the integral of skin squared of you minus 1, and that is going to integrate to be tangent of you minus you. So, this will allow us to simplify the integration by parts to be U multiplied by tangent squared of you, minus the quantity, tangent of U minus U, which will be negative tangent of you plus Cu. And because we were working with an indefinite integral, we will add a generic constant C. So what we need to do now is we need to resubstitute all the terms of you back in terms of X. Now, every U here is going to substitute back to tangent inverse of X. We know that tangent squared of you is going to simplify to be X, but what about tangent of you? Well, if we were to take the square root of both sides of this X statement, then we know that tangent of U is equal to the square root of X. And so by back substituting, we can rewrite the integration of parts to be tangent inverse. Of the square root of X. Multiplied by X minus the square root of X. Plus tangent inverse of the square root of X plus C. Now, if we were to group up, The tangent inverse terms and factor out the tangent inverse, we can further rewrite this to B X + 1 multiplied by tangent inverse of the square root of X. Minus the square root of X plus C, and this is going to be the simplest form of the solution to the given integral, and the overall solution is going to be B. So I hope this video helps you in understanding how to approach this type of integral, and we will go ahead and see you all in the next video.

92–98. Evaluate the following integrals.
97. ∫ tan⁻¹(∛x) dx

Key Concepts

Integration

Inverse Functions

Chain Rule

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