Find the integral by substitution: integral {16 x^3} / {x^4 + 5} dx.

Question:

Find the integral by substitution:

{eq}\displaystyle \int \dfrac {16 x^3} {x^4 + 5}\ dx {/eq}.

U-Substitution:

Integration can be done on functions in multiple methods. We can find the integral of some functions with the help of substitution. This method makes use of substitution on a term in the integrand, which corresponds to the change in the differential term. An appropriately-performed substitution results in an integration process that is easy to evaluate.

Answer and Explanation:

We allow the substitution, {eq}\displaystyle u = x^4+5 {/eq}, which correponds to a change in the differential term, {eq}\displaystyle du = 4x^3dx {/eq}. We apply the derivative and then apply the integral. We proceed with the solution.

{eq}\begin{align} \displaystyle \int \frac{16x^3}{x^4+5}dx &= \int \frac{4}{u}dx\\ &= 4\ln u +C\\ \text{We revert the substitution.}\\ &= 4\ln (x^4+5) +C \end{align} {/eq}


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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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