Choose the substitution function u(x)=x2+1 and apply the substitution method to calculate ∫xx2+1dx You may answer intermediate steps to check if you are on the right track.
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Step 1: Identify the substitution function \( u(x) = x^2 + 1 \). Show more…
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Suppose that you want to re-write an integral using a substitution, in this case, ∫ (x^5 / √(1-x^3)) dx = -1/3 ∫ ((1-u) / √ u) du Determine the correct substitution that will accomplish this. That is, find u as a function of x that allows you to re-write the integral as shown above. The function u(x) we want is u = 1-x^3 in which case the differential of u is du = -3x^2dx Note: answer should be in the form u = f(x) and du = f'(x)dx Part 2. Evaluate the indefinite integral above in terms of u. -1/3 ∫ ((1-u) / √ u) du = Note: answer should be in terms of u only Part 3. Back substituting in the antiderivative you found in Part 2. above we have ∫ (x^5 / √(1-x^3)) dx = -2/9(x^3+2)(sqrt(1-x^3)) Note: answer should be in terms of x only
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Use the substitution u = x^2 + 1 to evaluate the indefinite integral below:
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To apply substitution to an integral, the solver begins with a guess. The solver guesses that by changing the variable (here x) to u=g(x), the integral can be transformed. However, not every guess leads to a proper substitution. Consider the following integral. ∫(x² + 2)¹² dx Does the substitution u = x² + 2 apply? If so, you should be able to complete the integration.
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