# Determine the following indefinite integral. {eq}\displaystyle \int 7 x e^{\displaystyle 3x^2 + 2}\ dx {/eq}

## Question:

Determine the following indefinite integral.

{eq}\displaystyle \int 7 x e^{\displaystyle 3x^2 + 2}\ dx {/eq}

## Integration:

Calculating an integral is called integration. Mathematicians utilise integrals to determine a variety of useful quantities, including areas, volumes, displacement, etc. For antiderivatives, indefinite integrals are utilised. When we discuss integrals, we often refer to definite integrals.

## Answer and Explanation: 1

Given:

• We have given the function as {eq}7x{{e}^{3{{x}^{2}}+2}}{/eq}.

• The objective is to find the integration of given function.

Now,

{eq}\int{7x{{e}^{3{{x}^{2}}+2}}dx}=7\int{x{{e}^{3{{x}^{2}}+2}}dx}{/eq}

Suppose,

{eq}\begin{align*} {{x}^{2}}&=u \\ 2xdx&=du \end{align*}{/eq}

Now, substitute this value in given integral,

{eq}\begin{align*} 7\int{x{{e}^{3{{x}^{2}}+2}}dx}&=7\int{{{e}^{3u+2}}\frac{du}{2}} \\ & =\frac{7}{2}\int{{{e}^{3u+2}}du} \end{align*}{/eq}

Now,

{eq}\begin{align*} \frac{7}{2}\int{{{e}^{3u+2}}du}&=\frac{7}{2}\left( \frac{{{e}^{3u+2}}}{3} \right)+C \\ & =\frac{7}{6}{{e}^{3u+2}}+C \\ & =\frac{7}{6}{{e}^{3{{x}^{2}}+2}}+C \end{align*}{/eq}

Hence, {eq}\int{7x{{e}^{3{{x}^{2}}+2}}dx}=\frac{7}{6}{{e}^{3{{x}^{2}}+2}}+C{/eq}.