# Evaluate the integral using the given substitutions. integral {(3 x^2 + 10 x) dx} / {x^3 + 5 x^2...

## Question:

Evaluate the integral using the given substitutions.

{eq}\displaystyle \int \dfrac {(3 x^2 + 10 x)\ dx} {x^3 + 5 x^2 + 18 } {/eq}, substitution {eq}u = x^3 + 5 x^2 + 18 {/eq}.

## Integration By Substitution:

Integration by substitution is a method to solve the integral which is set up in the form of {eq}\int f(g(x))g'(x)dx {/eq}.

In this method, we transform the given integral equation into another integral form by making the substitution {eq}g(x)=t {/eq} and after evaluating the integral in terms of {eq}t {/eq}, reverse the substitution to get the required result in terms of {eq}x {/eq}.

Essential formula:

{eq}\int \dfrac{dt}{t}=\ln \left | u \right |+C {/eq}

Given

{eq}\displaystyle \int \dfrac{(3 x^2 + 10 x)\ dx}{x^3 + 5 x^2 + 18 } {/eq}

We have to solve the given integral.

We have to use the substitution {eq}x^3 + 5 x^2 + 18 =t {/eq} to solve the integral.

Differentiate both sides of the substitution with respect to {eq}x {/eq}

{eq}\ \begin{align} \dfrac{\mathrm{d} }{\mathrm{d} x}\left ( x^3 + 5 x^2 + 18 \right ) &=\dfrac{\mathrm{d} t}{\mathrm{d} x}\\ 3x^2+5(2x)+0 &=\dfrac{\mathrm{d} t}{\mathrm{d} x}\\ 3x^2+10x &=\dfrac{\mathrm{d} t}{\mathrm{d} x}\\ (3x^2+10x)dx &= dt\\ \end{align} {/eq}

Applying this substitution to the integral, we have:

{eq}\begin{align} \int \dfrac{(3 x^2 + 10 x)\ dx}{x^3 + 5 x^2 + 18 } &=\int \frac{dt}{t}\\ &=\ln \left | t \right |+C\\ \end{align} {/eq}

Reversing the substitution, we have:

{eq}\color{blue}{\int \dfrac{(3 x^2 + 10 x)\ dx}{x^3 + 5 x^2 + 18 }=\ln \left | x^3 + 5 x^2 + 18 \right |+C}\\ {/eq} 