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.

∫ dx / (x² √(4x - 9))

Accepted Answer

Hello. In this video, we are going to be integrating the following integral by using the standard integral table. Now, the integral given to us is the integral of 1 divided by X2 multiplied by the square root of 5 X minus 7. Now by the table of integrals, we can apply an identity to this first integral. We have an identity that states that if we have an integral of the form 1 divided by x2 multiplied by the square root of a x + b, then this is going to integrate to be the negative square root of a x plus b divided by b x. Minus a divided by 2b multiplied by the integral of 1 divided by x multiplied by the square root of a x plus b. Now with our given integral, our value of a is going to be 5 and our value of B is going to be -7 and by using this identity we can go ahead and simplify this first integral to be the quantity positive square root 5 x minus 7 divided by 7 x plus 5 divided by 14. Multiplied by the integral of 1 divided by x multiplied by the square root of 5 x minus 7. Now what we can go ahead and do at this step is we can use another identity to evaluate the 2nd integral that we obtained by the 1st identity. Now the identity for the 2nd integral is the following. If we have an integral of the form 1 divided by x multiplied by the square root of a x plus b, then this integral will evaluate to be 2 divided by the square root of b multiplied by tangent inverse of the quantity square root a x minus b divided by b. Now here, our value of A is going to be 5, and our value of B is going to be 7. And so by applying the identity, well, we will first write down our first term, that is going to be the square root of 5 x minus 7 divided by 7 x plus 5 divided by 14 multiplied by the quantity which we get from the identity. That is going to be 2 divided by the square root of 7 multiplied by the tangent inverse of the quantity square root 5 x minus 7 divided by 7. And then, because this is the final. Final solution, we will add plus C at the end. Now, what we can go ahead and do is we can multiply 5 divided by 14 and 2 divided by the square root of 7 together. And so that's going to leave us with the square root of 5 X minus 7 divided by 7 X plus 5 multiplied by the square root of 7 divided by 49, multiplied by the tangent inverse of the square root, 5 X minus 7 divided by 7 plus C. This is going to be the simplest form of the integral by using the standard integral table, and with that being said, the overall solution to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.∫ dx / (x² √(4x - 9))

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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x² √(4x - 9))