7-56. Trigonometric substitutions Evaluate the following integ...

Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

16. ∫ x²/(25 + x²)² dx

Accepted Answer

Determine the indefinite integral by performing a trigonometric substitution, where we have the integral of X2 divided by 16 plus X2 squared X. Now to solve this, we first need our substitution. In our denominator, we have the form of A2 plus X2. This tells us X should be a tangent of data. In this case, A will be 4, which means X will be 4, tangent of theta. DX then will be 4, C square theta, D theta. Let's go ahead and substitute these values into our integral. We have the integral Of 4 tangent theta squared. Divided by 16 plus. Or tangent data. When? All squared Multiplied by DX, which in our case, is 4s can square the D theta. Let's go ahead and simplify. We end up getting 16. 10 square data on the top. Divided by 16 plus 1610 square data. All squared on the bottom. Multiplied by 4, seek it squared theta. Do that. We can further simplify by using our identity for tangent square. We can write this then As the integral of 16. Tianet Square Theater. Divided by the denominator which is now. 256, sequent to the 4th, theta. We can expand that By changing 16 plus 16 tangent square theta into 16 multiplied. By 1 plus tangent square data, or 16 sequence square data. You would then multiply that. Bye Or skin square data. The the Let's go ahead and simplify a little bit more. We end up then getting 16 Ted at Square Theater. Multiplied By 4, divided by 256 sequence square theta when we cancel out our sequent squared. We'd like to get 64. Tendent Square Theater. Divided by 256 sequin square data, which finally simplifies. To 1/4. And then sin squared data. He did that. Now, let's go ahead and take our antiderivative. Now we have to change sign squared first, using our power of reduction identity. Will write this as the integral of 1/4th multiplied. By our power reduction identity, which is 1 minus cosine 2 theta, divided by 2. We can then simplify this to get 1/8 multiplied by the integral of 1 minus cosine to data. Now, let's go ahead and integrate this. This will be 1/8 multiplied by theta. Minus One half sin to the. Plus C. Now, let's go ahead and substitute some values. Now we first remember That we had Tangent In our case, we had 8. A theater So we have X equals 4 tangent data. Or tangent data equals x divided by 4. Giving us data equals the tangent inverse of X divided by 4. Now, we also notice, we have sin to data, which is a double angle identity. We have 1/8 multiplied by theta minus 1/2, 2, sine theta, cosine theta. Or 1/8 multiplied by theta minus sine theta, cosine theta. Let's see. Now we just need to find sine theta and cosine theta. We can go ahead and draw a triangle to determine this. We have a triangle. With theater. And we know we have tangent data equals x divided by 4. That means our opposite side will be X, adjacent side will be 4. All news then will be the square root of 16 + X2. This means then we can find sign data. Which will be X divided by the square root of 16 + X2. And cosine theta, which shall be 4, divided by the square root of 16 plus X2. We can now write our final integral. 1/8 multiplied. By inverse tangent, X divided by 4. Minus X divided by the square root of 16 plus X2. Multiplied By 4 divided by the square root of 16 plus X2. We can finally simplify once more to get 1/18. Change inverse x divided by 4. Minas Or X divided by 16 plus X squared, all plus C. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
16. ∫ x²/(25 + x²)² dx

Key Concepts

Trigonometric Substitution

Integral of Rational Functions

Pythagorean Identity

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