Integrate\int \frac{x+3}{x \sqrt{x^{2}+6x}}dx


Integrate {eq}\int \frac{x+3}{\sqrt{x^{2}+6x}}dx {/eq}

Substitution Rule

We can perform a u-substitution to approximate integrals containing a function and its derivative. In this case, we assign an expression to u, and using derivatives, exchange dx for du. This derives directly from the chain rule and relates integrals to derivatives through antidifferentiation.

Answer and Explanation:

Here, we choose

{eq}\displaystyle u = x^2 + 6x {/eq}.

If we do this, we see that

{eq}\displaystyle \begin{align*} \frac{du}{dx} &= 2x + 6 \\ \frac{du}{2} &= (x + 3) \ dx \end{align*} {/eq}

Then we obtain

{eq}\displaystyle \begin{align*} \int \frac{x+3}{\sqrt{x^{2}+6x}}dx &= \int \frac{du}{2\sqrt{u}} \\ &= \frac{1}{2}\int u^{-1/2} \ du \\ &= u^{1/2} + C \\ &= \sqrt{x^2 + 6x} + C \end{align*} {/eq}

Learn more about this topic:

How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5

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