37–56. Integrals Evaluate each integral.

∫₂₅²...

Question

37–56. Integrals Evaluate each integral.

∫₂₅²²⁵ dx / (x² + 25x) (Hint: √(x² + 25x) = √x √(x + 25).)

Accepted Answer

Evaluate the integral from 9 to 81 of DX divided by the square root of X2 + 9 X. Now, to solve this, let's modify our integral. This first factor X2 + 9 X. We can factor this as X multiplied by X plus 9. Now, what we can do is we can choose our substitution. Let's let you Equals the square root of X plus 9. This means you squared equals x plus 9. And X equals u2 minus 9. DX then will be 2 UDU. Let's go ahead And replace Our substitutions and are integral now. We know Have DX which will change. Into one divided by On the bottom, we get U square roots of U2 minus 9. Multiplied By 2 U D U. This gives us 2 U divided by U square root of U quad minus 9. Our uses cancel to give us 2 divided by the square root of U2 minus 9. Now, we're gonna take the integral of this, but we first need to find our new bounce. So let's take a look. We have U equals the square root of X plus 9. And so we have U equals the square root of 81 + 9. And the square root of 9 + 9. We get the square of 90 And the square root of 18. We can simplify these, we end up getting 3 squares of 10. For the square root of 90 And 3 squares of 2, with the square of 18. This replaces our bounces. Now, let's go ahead and solve our integral. We noticed then that we have an integral of a certain form. We have the integral of the form one divided by the square root. A few square minus A2. Which this integral resolves to the natural lag of the absolute value of u plus the square root of u square minus a 2 plus c. So, we can go ahead and change this for our integral. We end up getting two natural log of the absolute value. A view plus a square root of view squared minus 9. From 32 to 2 to 32 to 10. Then from here, we plug in our bounces. We'll end up kidding. Two values We end up getting the natural like. Of 3 square roots of 10. Plus 9 Minus the natural leg. Of 3 squares of 2. Plus 3. This is all multiplied by 2. We cannot simplify even further. We get 2 natural leg. Of 3 square of 10 plus 9, divided by 3 square of 2 + 3. Which can then simplify even further to two natural log. Of the square root of 10 + 3 divided by the square of 2 + 1. This is the final answer to our problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

37–56. Integrals Evaluate each integral.

∫₂₅²²⁵ dx / (x² + 25x) (Hint: √(x² + 25x) = √x √(x + 25).)

Key Concepts

Integration of Rational Functions

Partial Fraction Decomposition

Algebraic Manipulation and Substitution

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