Multiple substitutions If necessary, use two or more substitut...

Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.

∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)

Accepted Answer

Evaluate the integral of D theta divided by the square root of 2 plus the square root of 2 plus theta. And we're given a hint that's the start with U equals square root of 2 plus theta. Let's go ahead and solve this. Well, first, use our hint. U equals the square root of 2 plus theta. Let's go ahead and solve this for theta. We get U2 equals 2 + theta and theta equals U2 minus 2. We can then find the data, which will be 2UDU. We can now substitute this into our equation. We have the integral. Of to you do you. Divided by the square root of 2. Plus you Now that we have this, we can use another substitution. In this case, we will let V equal 2 plus U. Which means that you will equal V minus 2. We'll end up getting a new integral. Of 2 multiplied by the minus 2. Divided by the square root Of feet And this will be in terms of the the. Let's go ahead and solve this. We have 2 multiplied by the integral of V minus 2 divided by the square root of V. Which can be split up as 2 multiplied by the integral. A V divided by the square root of V. Minus 2 divided by the square of V. TV Now we split it up. And we can go ahead And use a substitution. For our exponents. In particular, we have 2 multiplied. By the integral A V to the 1/2. Minus 2 to the negative 1/2. T V We can now innovate this using our in-school Power rule. We have 2 multiplied. By the results of art in the world. Which in our case will be given by 2/3, so the three Fs. -2, multiply. By 25 to 1/2. We can simplify again. To get 43 be the three Fs. Minus or for the 1/2. And we also noticed we had a 2 inside of our parentheses from earlier, and so we can rewrite that once again to 8 of the 12. We will add a C at the ends. We can now simplify it once more. To get 45 to the 1 half power multiply. By the divided by 3. -2 by factoring out some turns. Now let's go ahead and use all of our substitutions again. Now, remember, we have V equals 2 plus U. Which means we will first have our substitution. You too plus you. And we will get 4 multiplied by 2 plus you. To the one half Multiplied By 2 plus U, divided by 3 minus 2 + C. And then we have our next substitution for a U, which we say U at the very beginning was a square root of 2 plus data. We get 4 multiplied by 2, plus the square root by 2 plus data. All raised to the one half power, multiplied by 2 plus the square root, 2 plus data. Divided by 3 minus 2. Plus C We can now simplify it once more. We will get 4 multiplied by 2, plus the square root of 2 plus data. To the one half power. Multipple by 2 plus the square root of 2 plus data. Minus 6, all divided by 3, by combining our two terms into one fraction. Finally, We have 2 minus 6, which will give us -4 in our second factor. Meaning we end up getting our final answer. We will get the final answer. A 4/3 What By 2, plus the square root of 2 plus data. So the one half, multiplied. By the square root of 2 + data. -4. All plus C. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.

  ∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)

Key Concepts

Substitution Method in Integration

Multiple Substitutions

Understanding Square Roots in Integrals

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