Problem 4

Question

Find the indefinite integral. $$ \int \frac{x^{2}}{3-x^{3}} d x $$

Step-by-Step Solution

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Answer

The indefinite integral of the function $ \frac{x^{2}}{3-x^{3}} $ with respect to $ x $ is $ - \frac{1}{3} \ln |3 - x^{3}| + c $

1Step 1: Substitute the function

Set a new variable $ u = 3 - x^3 $. Then calculate the differential of $ u $, which is $ du = -3x^{2} dx $. To isolate $ dx $, express it as $ dx = - \frac{du}{3x^2} $.

2Step 2: Replace in integral

Replace $ x^{2} dx $ with $ - \frac{du}{3} $ in the integral and the function $ 3 - x^{3} $ with $ u $. This turns the integral into $ - \frac{1}{3} \int \frac{1}{u} du $.

3Step 3: Simplify the integral

Now the integral is simplified to one of the elementary form, that can be directly integrated which results in $ - \frac{1}{3} \ln |u| + c $, where $ c $ denotes the constant of integration.

4Step 4: Back substitute $ u $ with $ x $

Now, replace $ u $ back by $ 3 - x^{3} $ from the Step 1. This results in $ - \frac{1}{3} \ln |3 - x^{3}| + c $.

Problem 4

Other exercises in this chapter

Problem 4

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Problem 4

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Problem 4

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of $n .$ Round your answers to four decimal

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Problem 4

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