Evaluate the integral \int (6x)e^{(3x^2+6)} \, dx using appropriate substitution.

Question:

Evaluate the integral {eq}\int (6x)e^{(3x^2+6)} \, dx {/eq} using appropriate substitution.

Integration by U-Substitution:

Let us apply the technique of u-substitution to find integral. This is because one of the coefficient terms is the differential coefficient of the other term. Therefore, we will substitute for terms in the exponent of the given exponential function. The following formula of integration is applicable:

{eq}\displaystyle\int e^x=e^x+C {/eq}

Answer and Explanation:

{eq}\begin{align} & \int (6x)e^{(3x^2+6)} \, dx \end{align} {/eq}

We will use the following substitution:

{eq}\begin{align} (3x^2+6)&=u\\ \Rightarrow 6x\, dx &=du\\ \end{align} {/eq}

Now, substitute these into the integral:

{eq}\begin{align} \int (6x)e^{(3x^2+6)} \, dx &=\int e^{(3x^2+6)} \, (6x) dx\\ &=\int e^{u} \, du\\ &= e^{u} +C\\ &= e^{(3x^2+6) } +C& (\text{Reverse the substitution } )\\ \end{align} {/eq}

Where {eq}C {/eq} is a constant of integration.

Therefore, the integral {eq}\displaystyle \boxed{\color{blue} { \int (6x)e^{(3x^2+6)} \, dx = e^{(3x^2+6) } +C }} {/eq}


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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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