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Find the integral: {eq}\int \frac{5x^2}{3x^3 + 2} \, \mathrm{d}x {/eq}.

Question:

Find the integral: {eq}\int \frac{5x^2}{3x^3 + 2} \, \mathrm{d}x {/eq}.

The Substitution Method

The integral of this problem has the form {eq}\int f(g(x)) g'(x)\, \mathrm{d}x {/eq}. This kind of integral can by solved applying the substitution {eq}u=g(x) {/eq} and {eq}du=g'(x)\, \mathrm{d}x {/eq}, with which this integral can be wriiten as:

{eq}\int f(g(x)) g'(x)\, \mathrm{d}x=\int f(u) \, \mathrm{d}u {/eq}

Answer and Explanation: 1

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Let {eq}u=3x^3 + 2 {/eq}, then {eq}\mathrm{d}u=9x2\, \mathrm{d}x {/eq}. By substitution,

{eq}\displaystyle \int \frac{5x^2}{3x^3 + 2} \,...

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How to Solve Integrals Using Substitution

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Chapter 13 / Lesson 5
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Explore the steps in integration by substitution. Learn the importance of integration with the chain rule and see the u-substitution formula with various examples.


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